Abstract
We establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for those equations in the elliptic case. The latter, which contains as main structural element a Cauchy kernel on the torus, is obtained from a dimensional reduction of the elliptic direct linearisation scheme of the lattice Kadomtsev–Petviashvili equation, which requires the introduction of a novel technical concept, namely the ‘elliptic cube root of unity’. Thus, in order to implement the reduction we define, more generally, the notion of elliptic \(N{\textrm{th}}\) root of unity, and discuss some of its properties in connection with a special class of elliptic addition formulae. As a particular concrete application we present the class of elliptic multi-soliton solutions of the lattice Boussinesq systems.
Similar content being viewed by others
1 Introduction
Many integrable equations, be it partial differential or difference equations, or autonomous ordinary differential equations or the difference equations describing integrable dynamical mappings, admit special solutions in terms of elliptic functions (or in terms of the associated quasi-periodic functions). A prototypical example is the Hirota equation
which is an integrable 3-dimensional lattice equation, i.e., a partial difference equation (P\(\varDelta \)E), with arbitrary fixed coefficients A, B, C for the function \(\tau _{n,m,l}\) of three discrete variables \(n,m,l\in {\mathbb {Z}}\). This equation allows for elliptic (type) solutions in the form
provided the coefficients take the form:
where \(\sigma (x)\) is the Weierstrass \(\sigma \)-function, and where \(\delta \), \(\varepsilon \) and \(\nu \) are fixed parameters, called lattice parameters, while \(\xi _0\) is an initial value. The fact that the Weierstrass \(\sigma \) function solves the Hirota bilinear equation is a direct consequence of the three-term addition formula for the latter function, (cf. Eq. (A.6) of Appendix A). Typically, this is not the only elliptic type solution of (1.1a), but a special solution from which infinite families of such solutions can be constructed involving determinants with elliptic entries, cf. e.g. [39]. In the continuous case of integrable PDEs, e.g. Korteweg–de Vries (KdV) or Boussinesq (BSQ) type equations, such elliptic solutions exist as well, e.g. the well-known cnoidal wave solution (in terms of the Jacobi cn-function) of the KdV equation, or in the case of ODEs the Lamé-Baker function solution of the Lamé equation.
In recent years the theory of integrable difference equations has grown into a substantive body of new insights, cf. e.g. [12]. In this context, certain classes of integrable partial difference equations (otherwise known as lattice equations) have gained prominence, e.g. the class of quadrilateral lattice equations integrable through the so-called multidimensional consistency (MDC) property. The scalar affine-linear family of such equations was classified by Adler, Bobenko and Suris (ABS) in [2] and subsequently their elliptic solutions were established in [5, 26]. Whereas the ABS equations are for single-component functions, elliptic solutions for higher rank equations, such as the lattice equations in the Gel’fand-Dikii (GD) hierarchy of [29] remained outstanding. A particular case of such equations is the (rank 3) lattice Boussinesq (BSQ) system, [29], which reads:
with P and Q being lattice parameters, and which can be visualised as a relation between the values of the solution function on the vertices on a 9-point stencil, or can be reformulated as a 2-component quadrilateral lattice system. The latter type of systems have yet to be classified, and our knowledge about both the structure of the equations (e.g. in terms of the polynomial structure of the function defining the equation, which is no longer affine-linear, as in the case of [2]) as well as the solution structure is as yet incomplete. In [29], cf. also [24, 25, 37], Eq. (1.2) and its companion equations (lattice modified BSQ equation and lattice Schwarzian BSQ equation), and their extensions (in the sense of the unfolding of the dispersion curves, cf. [40]) were constructed, and the integrability aspects (Lax pair, reductions, Poisson structure) were established, on the basis of a framework called direct linearisation (DL). The latter framework goes back to early papers by Fokas et al., [9, 33], and is based on singular linear integral equations over arbitrary contours in the complex space of the spectral parameter, and allows to formulate the solution of spectral problems and corresponding nonlinear evolution equations in a compact way. The DL idea was subsequently developed in [31] into a flexible, albeit formal, machinery involving infinite matrix structures, having the advantage that several (Miura-related) equations can be treated within one framework. Furthermore, it allows the treatment of discrete as well as continuous equations on one and the same footing and within one formalism. Furthermore, elliptic type solutions can be treated within the formalism by a suitable choice of elliptic Cauchy kernel. Although the DL approach deviates from the inverse scattering formalism in that it doesn’t address the issue of initial value problems, it nonetheless covers large classes of solutions, including soliton solutions and solutions can be otherwise obtained through the inverse scattering technique. However, to obtain algebra-geometric solutions the DL would have to be extended in a nontrivial way.
In the present paper we perform a first step towards the latter goal, namely to set up the framework for elliptic type solutions of the lattice BSQ systems. The reason why the BSQ system is of particular interest, is that, in spite of the interest that has been raised about the scalar quad-lattice equations of the ABS list, [2], they are still rather special and the corresponding techniques to study their integrability give little insight into what happens with multicomponent systems or lattice systems of higher order. Thus, is of interest to study elliptic solutions of the BSQ system, as it is the first case in the lattice GD hierarchy beyond the scalar quad-lattice situation, and quite representative of the generic case. The elliptic solutions we are interested in include both ‘elementary’ elliptic solutions, as well as multi-solitonic towers of elliptic solutions and in principle also the inverse scattering type solutions based on elliptic asymptotic behaviour. Thus, the paper extends to the higher rank case the results obtained in [26] for the ABS list (excluding the case of the so-called Q4 equation, which was covered in [5], but an analogue of which has not yet been established in the higher-rank caseFootnote 1). The strategy in the present paper, to set up the DL scheme is to realise the structure as a reduction of the one for the lattice Kadomtsev–Petviashvili (KP) system, whose elliptic type solutions were constructed in [39]. However, a technical problem in the BSQ case (which is one instance demonstrating that the quad-lattice is not representative of the generic situation) is the need for a new concept in the reduction. In fact, in the rational case, the reduction from KP to BSQ (or to other systems in the GD family) involves cube- or higher roots of unity. In order to perform the reduction from KP to such a system in the elliptic case, it turns out that an elliptic analogue of the roots of unity are needed. Such objects were considered only in the case \(N=3\) in [36] without explicitly referring to them as elliptic analogues of cube roots. Here, we introduce such objects for general N relying on higher-order elliptic identities which are given in Appendix B, (cf. also [7]). With the help of these elliptic roots of unity the reduction from the KP system to the BSQ and the higher-rank systems can be implemented via conditions on combinations of lattice shifts. Thus, from the DL structure for the lattice KP system we readily obtain the solution structure of the lattice GD systems, [29], within the DL framework.
The organisation of the paper is as follows. In Sect. 2 we develop the DL scheme for elliptic type solutions of the lattice KP class with the aim to make dimensional reductions to the KdV and BSQ cases. For the sake of the latter reduction we need an elliptic analogue of the cube root of unity. Thus, we introduce in Sect. 3 the notion of elliptic \(N{\textrm{th}}\) root of unity, and examine some of its properties. In Sect. 4 we employ this novel concept to set up the DL system for the elliptic type solutions for the BSQ lattice equations, while in Sect. 5 we derive the actual nonlinear equations in closed form. In Sect. 6 we present explicit examples of elliptic solutions, namely elliptic seed and elliptic multi-soliton solutions, for the equations in their ‘standard form’. Some conclusions follow in Sect. 7. For completeness we have given the elliptic parametrisation for the Lax pairs in Appendix D , although we do not need them for the purpose of presenting the elliptic solutions in this paper.
Warning Some standard facts on elliptic functions are introduced in Appendix A and we make heavy use of the various addition formulae throughout the paper, in particular some higher order addition formulae as given in Appendix C. The equations as they emerge from the structure are in our opinion best represented using suggestive notations, distinguishing variables from lattice parameters, but in the elliptic case both variables and parameters are defined through elliptic functions. Thus, as the story unfolds, we find it convenient to introduce several layers of somewhat ad-hoc notation in order to write the equations and the relations between them in what we feel is the most lucid way.
2 Elliptic DL Scheme for the Lattice KP Equations
In this section we develop the DL structure for elliptic type solutions of KP lattice equations. The structure is based on a unifying framework of formal linear integral equations, from which the various lattice KP equations can be derived. The set-up is that we start from a large solution class in terms of which a multitude of functions on the lattice can be defined, each of which solves an equation (or rather a parameter-family of equations) in the KP class. Thus, the DL structure not only provides solutions of the individual KP equations, but also supplies us with the (Miura/Bäcklund) relations between the various equations.
2.1 DL framework
In an early paper [27] we derived a system of integrable lattice equations (i.e. partial difference equations) associated with the lattice KP equation from a system of linear integral equations. In the present paper we generalise that integral equation to the elliptic case. The essential difference with the rational case is that the rational Cauchy kernel \((k+l')^{-1}\) in the integral equation of [27] is replaced by an elliptic kernel of the form \(\varPhi _\xi (\kappa +\ell ')\) where \(\varPhi \) is the Lamé type function
in which \(\sigma (x)\) is the Weierstrass sigma-function, [4]. This function has a number of remarkable properties, notably an elliptic analogue of the partial fraction expansion formula, namely Eq. (A.7) of Appendix A. Further properties of this and associated functions are given in Appendix A. An important difference between the elliptic and rational case is the introduction of the auxiliary complex variable \(\xi \) which plays the role of a background variable in all the formulae. In fact, when we characterise solutions from this scheme as being ‘elliptic’, it means that they are elliptic or quasi-elliptic functions of this variable \(\xi \). (Note that in what follows this variable is so far free and undetermined, but will be later specified as being non-autonomous, i.e., dependent on the discrete independent variables of the system of partial difference equations that we will derive.)
We will now define the DL framework by writing down a set of linear integral equations in which the elliptic Cauchy kernel given above is the main ingredient. They are linear integral equations for a vector-valued function \({{\varvec{u}}}_\kappa \), depending on a complex valued spectral parameter \(\kappa \), of the following formFootnote 2
and one for its adjoint vector \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\), depending on another spectral variable \(\kappa ^\prime \), of the form
The explanation of the notations in Eq. (2.2) requires some space. To start with the integrations, in both Eqs. (2.2a) and (2.2b) they are the same, and involve a general measure \(d\mu (\ell ,\ell ^\prime )\) and the integration domain is any suitably chosen subset \(D\subset {\mathbb {C}}^2\) in the space of both spectral variables \(\ell \) and \(\ell ^\prime \). At this point it is not necessary to specify the measure and integration domain, as we treat the integral equations on a purely formal level. We merely need to rely on some general assumptions (such as the assumption that for given measure and integration domain D the solution of the integral equation is unique). We note that the integral equation (2.2a), in the rational case where the kernel \(\varPhi _\xi (\kappa +\ell ')\) is replaced by \((\kappa +\ell ')^{-1}\), can be viewed as a generalisation of the singular integral equations appearing in the nonlocal Riemann-Hilbert (RH) problem or \({\bar{\partial }}\)-problem appearing in the inverse scattering approaches to KP type equations, cf. e.g. [10, 16]. However, those approaches start from a given multidimensional spectral problem (usually the time-dependent Schrödinger equation) and aim at solving initial value problems for (2+1)-dimensional KP type equations. Here our starting point is the integral equation itself, viewed as a formal structure that we aim to explore.
This brings us to the inhomogeneous term on the right hand sides of Eq. (2.2). These contain the (infinite) column and row vectors \(\varPhi _\xi ({\varvec{\varLambda }}) {{\varvec{c}}}_\kappa \) and \(\,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }\varPhi _\xi (\,^{t\!}{\varvec{\varLambda }})\), as well as the plane wave factors \(\rho _\kappa \) and \(\sigma _{\kappa '}\) (where the \(\sigma \) in \(\sigma _{\kappa ^\prime }\) should not be confused with the Weierstrass sigma-function) respectively which will need some introduction.
First, the plane wave factors \(\rho _\kappa \) and \(\sigma _{\kappa ^\prime }\) determine the dependence of the solutions \({{\varvec{u}}}_\kappa \) and \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\) on the main dynamical variables n, m, l, which can be assumed to take values in the integers, (i.e. \(n,m,l\in {\mathbb {Z}}\)), but in fact only need to increase in value by integer steps. In fact, we only need to assume that there is a set of elementary shift operations \(T_\delta \) which act on the plane wave factors as follows:
where the tilde is shorthand for the shift operation, and where the shift is characterised by a complex ‘lattice’ parameter \(\delta \). This lattice can be seen as the grid formed by multiple operations of the shift operator, and hence we can identify
Changing the lattice parameter \(\delta \) means adding another degree of shift freedom to the lattice, and effectively adding another independent variable to the system. Thus, in addition to (2.3), we can consider the simultaneous shift relations
characterised by lattice parameters \(\varepsilon \) and \(\nu \) respectively (the hat and bar again being shorthand for the operation of the shift). Since these shifts act on \(\rho _\kappa \) and \(\sigma _{\kappa ^\prime }\) by multiplicative factors, it is evident that these shift operations commute, i.e. \(T_\delta (T_\varepsilon \rho _\kappa )= T_\varepsilon (T_\delta \rho _\kappa )\), etc. (and similarly for their action on \(\sigma _{\kappa ^\prime }\)) and hence we can consider the simultaneous iteration of these shifts, creating a three-dimensional lattice of multiple shifts labelled by discrete variables (n, m, l). Thus, we create plane wave factors \(\rho _\kappa (n,m,l)\) and \(\sigma _{\kappa ^\prime }(n,m,l)\) which by construction become functions of the three-dimensional lattice, i.e.
relative to some initial values \(\rho _\kappa (0,0,0)\), \(\sigma _{\kappa '}(0,0,0)\). Thus, by construction we have
and similarly for the other lattice shifts:
Remark
Although n, m, l will play the role of the main independent variables for the partial difference equations which we will derive in due course, they are just parameters as far as the solutions of the integral equations (2.2) is concerned, and hence we assume that the measure \(d\mu (\ell ,\ell ^\prime )\) and integration domain D do not depend on these variables. Clearly, as a consequence of the dependence on the variables n, m, l of the plane wave factors, the solutions \({{\varvec{u}}}_\kappa \) and \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\) of the integral equations will acquire a dependence on these variables as well, and an aim of the DL approach is to unravel how that dependence will manifest itself in terms of shift relations for these objects.
Second, the (infinite) column and row vectors \(\varPhi _\xi ({\varvec{\varLambda }}) {{\varvec{c}}}_\kappa \) and \(\,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }\varPhi _\xi (\,^{t\!}{\varvec{\varLambda }})\) require some explanation. It was a finding going back to [31] that the dependence of the solutions of the integral equations on (continuous or discrete) dynamical variables is best controlled by considering an infinite set of integral equations with inhomogeneous terms given by different powers of the spectral parameter. This can be encoded in terms of the infinite (column resp. row) vectors \({{\varvec{c}}}_\kappa =(\kappa ^i)_{i\in {\mathbb {Z}}}\) and \(\,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }=(\kappa '^j)_{j\in {\mathbb {Z}}}\) of monomials in \(\kappa \) and \(\kappa '\) respectively, and the action of index-raising operators \({\varvec{\varLambda }}\) and \(\,^{t\!}{\varvec{\varLambda }}\), acting from the left and right respectively, as follows
In the rational case, from the integral equations like (2.2) one can then derive a set of fundamental shift relations in terms of these operators representing a set of infinite recurrence relations between the components of the vectors \({{\varvec{u}}}_\kappa \) and \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\), from which subsequently (by eliminating the operators \({\varvec{\varLambda }}\) and \(\,^{t\!}{\varvec{\varLambda }}\) combining different lattice shifts) the partial difference equations are derived in a systematic way, cf. e.g. [27, 29, 30]. In the elliptic case, however, due to the appearance of the elliptic Cauchy kernel, it is convenient to dress the inhomogenous terms with the factors \(\varPhi _\xi ({\varvec{\varLambda }})\) and \(\varPhi _\xi (\,^{t\!}{\varvec{\varLambda }})\), as is made evident from the analysis presented in Appendix B. This leads then to the notion of what one could call elliptic matrices: these are obtained by the formal substitutions of the index-raising operators in the arguments of the relevant elliptic functions, such as
where \(\wp (x)\) and \(\zeta (x)\) are the well known Weierstrass functions, with properties summarised in Appendix A. The introduction of such formal operators is justified by thinking of them in terms of their Fourier type symbols, which is the way in which they appear inside the integrals when performing computations. It is tempting to think of these operators as elliptic analogues of raising/lowering operators, defined through the formal power series expansions of the Weierstrass \(\sigma \), \(\zeta \) and \(\wp \) functions. Consequently, they are assumed to obey the same addition formulae as the usual elliptic functions (see Appendix A), but in terms of arguments containing the operators \({\varvec{\varLambda }}\) and \(\,^{t\!}{\varvec{\varLambda }}\). In terms of these elliptic matrices it is now possible to set up a formal algebraic structure, based on the integral equations (2.2), which allows us to derive the relevant nonlinear lattice equations as well as the associated Lax pairs.
2.2 Formal structure
Having explained the notations in Sect. 2.1, we can now set up and exploit the structure encoded by means of the index raising operators \({\varvec{\varLambda }}\) and \(\,^{t\!}{\varvec{\varLambda }}\), and reformulate the integral equations in a more formal way for the convenience of deriving the lattice systems hidden in the structure.
In fact, the formal DL scheme can be summarised as follows. It contains three types of ingredients:
-
An infinite ( \({{\mathbb {Z}}}\times {{\mathbb {Z}}}\)) matrix \({{\varvec{C}}}\) which we can take of the form
$$\begin{aligned} {{\varvec{C}}}=\iint _{D} d\mu (\ell ,\ell ^\prime )\,\rho _\ell \sigma _{\ell ^\prime }{\varvec{c}}_{\ell } \,^{t\!}{{\varvec{c}}}_{\ell ^\prime }\ , \end{aligned}$$(2.6)in which \({{\varvec{c}}}_\ell \) and \(\,^{t\!}{{\varvec{c}}}_{\ell }\) are infinite vectors with components \(({{\varvec{c}}}_\ell )_j=(\,^{t\!}{{\varvec{c}}}_{\ell })_j=\ell ^j\), and \(\rho _\ell \) depends on additional variables that are to be determined later. The integrations over region D and measure \(d \mu \) need not be specified at this point but we will loosely assume that they can be chosen such that the objects to be introduced below are well-defined.
-
The matrices \({\varvec{\varLambda }}\) and \(\,^{t\!}{\varvec{\varLambda }}\), as explained above, define the operations of index-raising when multiplied at the left respectively the right, acting on \({{\varvec{c}}}_\kappa \) respectively \(\,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }\) by \({\varvec{\varLambda }}{{\varvec{c}}}_\kappa =\kappa {{\varvec{c}}}_\kappa \) , resp. \(\,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }\,^{t\!}{\varvec{\varLambda }}=\kappa ^\prime \,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }\). Hence, these are right and left eigenvectors of the index raising operators.
-
A formal elliptic Cauchy kernel, i.e., an infinite matrix \({\varvec{\varOmega }}_\xi \) obeying the equations:
$$\begin{aligned} {\varvec{\varOmega }}_\xi \varPhi _\gamma ({\varvec{\varLambda }}) - \varPhi _\gamma (-\,^{t\!}{\varvec{\varLambda }}){\varvec{\varOmega }}_{\xi +\gamma } = \varPhi _\xi (\,^{t\!}{\varvec{\varLambda }}){{\varvec{O}}}\varPhi _{\xi +\gamma }({\varvec{\varLambda }})\ , \end{aligned}$$(2.7)in which \({{\varvec{O}}}\) is the projection matrix on the central element, i.e. \(({{\varvec{O}}} {{\varvec{C}}})_{i,j}=\delta _{i,0}{{\varvec{C}}}_{0,j}\), etc.
As a consequence of the shift relations (2.3) of the plane wave factors \(\rho _k\) and \(\sigma _{\kappa ^\prime }\) under translations along the lattice, we have the following linear relations for \({{\varvec{C}}}\):
for the various lattice directions associated with the lattice parameters \(p=\zeta (\delta )\), \(q=\zeta (\varepsilon )\), resp. \(r=\zeta (\nu )\). In principle, since these relations are linear, we can select an arbitrary set of lattice parameters p, q or r (or equivalently \(\delta \), \(\varepsilon \) and \(\nu \)) each associated with its own lattice shift, and in view of the linearity of the corresponding equations for \({{\varvec{C}}}\) we can impose all the discrete evolutions for all chosen values of these lattice parameters simultaneously.
The main objects of interest, however, are the ones obeying nonlinear equations, and these objects are the following:
-
The infinite matrix
$$\begin{aligned} {{\varvec{U}}}_\xi := \varPhi _\xi ({\varvec{\varLambda }}){{\varvec{C}}} \left( {{\varvec{1}}} + {\varvec{\varOmega }}_\xi {{\varvec{C}}}\right) ^{-1}\varPhi _\xi (\,^{t\!}{\varvec{\varLambda }})\ . \end{aligned}$$(2.9) -
The \(\tau \)-function given by the infinite determinantFootnote 3
$$\begin{aligned} \tau _\xi := \det \,_{{{\mathbb {Z}}}\times {{\mathbb {Z}}}}\left( {{\varvec{1}}}+{\varvec{\varOmega }}_\xi \cdot {{\varvec{C}}}\right) \ . \end{aligned}$$(2.10) -
The ‘wave vectors’ \({{\varvec{u}}}_\kappa \) and \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\) defined by
$$\begin{aligned} {{\varvec{u}}}_\kappa (\xi )= & {} \left( \varPhi _\xi ({\varvec{\varLambda }}) - {{\varvec{U}}}_\xi \varPhi _\xi ^{-1}(\,^{t\!}{\varvec{\varLambda }}) {\varvec{\varOmega }}_\xi \right) {{\varvec{c}}}_\kappa \rho _\kappa \ , \end{aligned}$$(2.11a)$$\begin{aligned} \,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }(\xi )= & {} \sigma _{\kappa '}\,^{t\!}{{\varvec{c}}}_{\kappa ^\prime }\left( \varPhi _\xi (\,^{t\!}{\varvec{\varLambda }}) - {\varvec{\varOmega }}_\xi \varPhi _\xi ^{-1}({\varvec{\varLambda }}) {{\varvec{U}}}_\xi \right) \ , \end{aligned}$$(2.11b)which can be identified with the solutions of the integral equations (2.2a) resp. (2.2b).
We note that the infinite-component matrix \({{\varvec{U}}}_\xi \) (2.9), with entries \(u_{i,j}(\xi )\), (\(i,j\in {{\mathbb {Z}}}\)), which are the main quantities of interest in what follows, also has the integral representation
(For notational convenience, here and elsewhere we indicate the dependence on the variable \(\xi \) by a suffix, rather than writing it as an argument, e.g. rather than writing \({{\varvec{U}}}(\xi )\).)
Let us now present the basic equations resulting from this scheme. From the above ingredients it is elementary to derive the basic relations describing the behaviour of \(\tau _\xi \), \({{\varvec{U}}}_\xi \) and \({{\varvec{u}}}_\kappa \) and \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\) under lattice translations. Thus, for the \(\tau \)-function we have (see Appendix B for the derivation):
and similarly for the other lattice directions. Subsequently we will adopt the shorthand notation where the infinite unit matrix symbol \({{\varvec{1}}}\) will be omitted when it multiplies a scalar function.
For \({{\varvec{U}}}_\xi \) we can derive the discrete matrix Riccati type of relations
Equation (2.14) together with its counterparts for the other lattice directions forms the starting point for the construction of a number of integrable three-dimensional lattice equations. By combining the different lattice translations associated with the different lattice parameters \(\delta \), \(\varepsilon \), \(\nu \) one can actually derive all relevant discrete equations within the KP family, as we shall show in the next section. Let us finish here by giving the linear equations for \({{\varvec{u}}}_\kappa \) and \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }\) that form the basis for the derivation of the Lax pairs for the above-mentioned equations derived from (2.14). These relations read
Equations (2.13), (2.14) and (2.15), the derivation of which is given in Appendix B, form the starting point for the subsequent derivation of nonlinear equations in closed form. For this purpose we have to specify special elements (or combinations of elements) of the infinite matrix \({{\varvec{U}}}_\xi \) and to combine the different lattice shifts in such a way that closed-form equations for such these preferred elements are obtained. This we will do in the next subsection.
2.3 Lattice KP systems from elliptic DL
We will now derive closed-form difference equations for special quantities defined in terms of the entries of the matrix \({{\varvec{U}}}_\xi \). First, combining the three relations of the form (2.14) for the three lattice directions and their shifted counterparts we can eliminate all the elliptic index raising matrices \(\zeta ({\varvec{\varLambda }})\) and \(\zeta (\,^{t\!}{\varvec{\varLambda }})\) and by taking the (0, 0) entry of the matrix, i.e. setting \(u(\xi ):= ({{\varvec{U}}}_\xi )_{0,0}\) , we get a closed form relation for the function \(u(\xi )\) which reads
in which ‘+ cycl.’ means the addition of similar terms obtained by cyclic permutation of the action of the shifts \(u\mapsto \widetilde{u}\), \(u\mapsto \widehat{u}\), \(u\mapsto \overline{u}\) together with corresponding replacements of the three parameters \(\delta \), \(\varepsilon \) and \(\nu \) respectively. Below we will identify (2.16) with the lattice potential KP equation of [27].
Further equations can be derived from (2.14) and its counterpart in the other lattice directions for the parameter-dependent quantities
for which from (2.14) we immediately have
which are motivated by the \(\tau \)-function relation (2.13) replacing the lattice parameter \(\delta \) by an arbitrary fixed parameter \(\alpha \). By eliminating the terms with \(\zeta (\,^{t\!}{\varvec{\varLambda }})\) in (2.14) and its counterpart with lattice parameter \(\varepsilon \) we can derive, for any fixed parameter \(\alpha \), the following relation
relating the lattice potential KP variable \(u(\xi )\) to the variable \(v_\alpha (\xi )\). To derive Eq. (2.19) use has been made of a special elliptic relation, namely Eq. (A.13) given in Appendix A. Using Eq. (2.19) for each pair of three lattice directions we can eliminate either \(v_\alpha (\xi )\), in which case we recover the lattice equation (2.16) for \(u(\xi )\), or we eliminate \(u(\xi )\) and obtain a lattice equation for \(v_\alpha (\xi )\), namely
This equation can be identified with the lattice (potential) modified KP (mKP) equation, cf [27].
By setting \(\alpha \) equal to either one of the three parameters \(\delta \), \(\varepsilon \) or \(\nu \), (2.20) reduces to a four-term equation and using the identifications (which follow from (2.13))
and similarly for the other lattice directions, we then obtain the Hirota bilinear KP equation in the following form
The latter coincides with (1.1a) with the coefficients given by (1.1b) using the relation (A.4) of Appendix A.
Another object of interest within the scheme is the following
for which we can derive from (2.14) the following important identity
and similar relations for the other lattice shifts \(\widehat{~}\) and 
involving the parameters \(\varepsilon \) and \(\nu \) instead of \(\delta \). In the derivation of (2.24) use has been made again of the special relation (A.13) of Appendix A. Eliminating the variables \(v_\alpha \) and \(w_\beta \) by combining three lattice shifts, we arrive at a closed-form equation for \(s_{\alpha ,\beta }\) which reads
in which we have used the abbreviation
For arbitrary \(\xi \) the Eqs. (2.16), (2.20) and (2.25) form a complicated system of non-autonomous difference equations for effectively a set of four-variable functions, depending on the discrete variables n, m, l and a continuous variable \(\xi \), as partial delay-difference equations. However, if we allow the variable \(\xi \) to depend on the discrete variables as follows:
where \(\xi _0\) is fixed, i.e., the \(\xi \) becomes dynamical, then those equations are nothing else than the lattice KP, lattice modified KP and lattice Schwarzian KP equations, cf. [27], respectively, and can be transformed into their standard forms by an appropriate gauge transformation, thereby removing the elliptic coefficients. We will not do so here, as they are not the main objects of study in the present paper, but explicit soliton type solutions were given along that line in [39]. We now consider the problem of their dimensional reduction to equations in the GD hierarchy, which in the elliptic case, requires the notion of the elliptic \(N{\textrm{th}}\) root of unity.
3 Elliptic \(N{\textrm{th}}\) Root of Unity
In the rational case, to perform the dimensional reduction from three-dimensional KP systems to two-dimensional lattice equations of GD type (including KdV or BSQ type equations), cf. [11, 29], we need to choose integration measures of the integral equations (2.2) where the spectral variable \(\ell '\) is identified with \(-\omega ^j \ell \), where \(\omega \) is a primitive \(N{\textrm{th}}\) root of unity, and \(j~(0<j<N)\) are integers coprime to N. However, this prescription no longer works, for \(N>2\), in the elliptic case, and in fact we need to revisit our notion of what is meant by the \(N{\textrm{th}}\) root of unity to get a sensible reduction. The main problem is that the multiplication by roots of unity within the arguments of the elliptic functions is not a natural operation and doesn’t work through naturally in the formulae for the reduction.
For \(N=2\), however, the usual square root of unity ‘\(-1\)’ can still be implemented within the arguments of the elliptic functions without a problem, but we need to reexamine what we actually mean by this elliptic root. In this context, we redefine it as follows: The elliptic square roots of unity are given by the condition
where \(\omega (\delta )\) is required to be independent of \(\kappa \). The condition (3.1) obviously has the simple solution \(\omega (\delta )=-\delta \), which coincides with what one has in the usual case of a square root of unity. Note that in this case the elliptic square root \(-\delta \) can be ‘normalised’ by dividing by \(\delta \), and hence we can extract a notion of square root (namely \(-1\)) which is independent of the parameter \(\delta \). However, this is no longer the case when \(N>2\), and in those cases the elliptic roots will essentially depend on a parameter.
Thus already for \(N=3\) we see an essential change of this notion, when we define the elliptic cube root of unity as follows:
Definition 3.1
We call the parameter-dependent quantities \(\omega _j(\delta )\) (\(j=0,1,2\)), where \(\omega _0(\delta )=\delta \), the elliptic cube roots of unity, if they obey the following relation
(for all \(\kappa \) in a fundamental domain of the argument of the elliptic functions) subject to the condition that the \(\omega _j(\delta )\), \(j=0,1,2\), are independent of \(\kappa \).
Assuming the existence of these elliptic cube roots,Footnote 4 which should not be confused with the half periods of the elliptic functions as in Appendix A, and which are determined modulo the period lattice, the following assertions can be made about the elliptic cube roots.
Lemma 3.1
The condition (3.2) is equivalent to the following set of relations:
Furthermore, we also have
Proof
Using (A.5) of Appendix A, we have that the left-hand side of (3.2) can be written as
whereas the right-hand side of (3.2), using (A.8) and (A.9), can be written as
Since, by definition, the left-hand side equals the right-hand side for all \(\kappa \) in a fundamental domain of the period lattice, we must have that \(\omega _1(\delta )+\omega _2(\delta )\equiv -\delta (\text {mod period lattice})\), i.e. Eq. (3.3a) must hold, and furthermore we have that (3.3b) is satisfied. Then from the latter we have from (A.9) that (3.3c) holds. Finally, from (A.10a) we subsequently have (3.3d). Conversely, from (3.3a) and (3.3b), using (A.5) we have
which is (3.2) holding for all \(\kappa \) in a fundamental domain. \(\square \)
Remark
From Lemma 3.1 it follows that we could have defined the elliptic root of unity by the relation (3.3c) subject to (3.3a). In fact, this point of view asserts the existence of the elliptic cube root of unity as follows:
Lemma 3.2
There exist solutions \(\omega _j(\delta )\) (\(j=0,1,2\)) of the relations (3.3c) subject to the relation (3.3a).
Proof
Since \(\wp '\) is an elliptic function with a pole of order three at zero, the equation \(\wp '(\omega )-\wp '(\delta )=0\), for any given \(\delta \), has, apart from the trivial solution \(\omega \equiv \delta (\textrm{mod period lattice})\), two independent solutions \(\omega _1,\omega _2\) modulo the period lattice of the elliptic function, i.e. solutions such that \(\wp (\omega _1)\ne \wp (\omega _2)\ne \wp (\delta )\). Furthermore, from the Frobenius-Stickelberger formula (A.11) it follows that
since the determinant on the left-hand side vanishes if (3.3c) holds. Hence, these roots must satisfy (3.3a) since \(\sigma (\delta +\omega _1+\omega _2)=0\) if \(\omega _j=\omega _j(\delta )\), (\(j=1,2\)). \(\square \)
Remark
The rational analogue of the relation (3.2) is the requirement that the third order polynomial of an indeterminate k
is a pure cubic in k, which leads to the solution that the roots \(p_0\), \(p_1\), \(p_2\) are related through cube roots of unity: \(p_j=\omega ^j p~ (j=0,1,2)\) with \(\omega =\exp \left( 2\pi \textrm{i}/3\right) \) (up to permutations of the roots) and \(\textrm{i}\) is the imaginary unit. Thus, the solutions of (3.2) as the elliptic analogue of a pure cubic and hence we can think of \(\omega _1(\delta )\) and \(\omega _2(\delta )\) as generalisations of \(e^{2\pi \textrm{i}/3}\delta \) and \(e^{4\pi \textrm{i}/3}\delta \), but here they are defined by the implicit equation (3.2). (In fact, when the invariant of the elliptic curve \(g_2=0\) the elliptic cube roots correspond exactly to these expressions in terms of the usual cube roots.) Note also that when \(\omega _1\) and \(\omega _2\) correspond to the half periods of the period lattice, i.e. when \(\omega _1(\delta )=\omega \) and \(\omega _2(\delta )=\omega '\), and \(\delta =-\omega -\omega '\) then the above relations (3.3) are all automatically satisfied by means of elliptic identities, since \(\wp '\) evaluated at the half periods vanishes: \(\wp '(\omega )=\wp '(\omega ')=0\).
In the case of \(N=4\) the elliptic quartic root of unity can be defined using the identity (C.2) of Appendix C, noting that here we need higher-order elliptic identities.
Definition 3.2
We call the parameter-dependent quantities \(\omega _j(\delta )\) (\(j=0,1,2,3\)), where \(\omega _0(\delta )=\delta \), the elliptic quartic roots of unity, if they obey the following relation
(for all \(\kappa \) in a fundamental domain of the argument of the elliptic functions) subject to the condition that the \(\omega _j(\delta )\), \(j=0,1,2,3\), are independent of \(\kappa \).
Once again the relation (3.4) gives rise to some properties of the elliptic quartic roots of unity, namely
Lemma 3.3
The condition (3.4) is equivalent to the following set of relations:
Furthermore, we also have
Proof
On the one hand, using the relation (C.2) of Appendix C the left-hand side of (3.4) can be written as
on the other hand, rewriting the right-hand side of (3.4), and using the relation \(\wp ''=6\wp ^2-g_2/2\), we have
which by comparing with the result of the left-hand side, given that \(\kappa \) is arbitrary, leads to the relations (3.5a)–(3.5c). Furthermore, (3.5d) follows from the substitution \(\kappa =-\omega _j\) into (3.4). Finally, Eq. (3.6) follows from (C.7) of Appendix C, setting the arguments equal to the elliptic quartic roots of unity. \(\square \)
As in the case of \(N=3\) we can use Eq. (3.5d) to define the quartic roots of unity, with the existence of the following an analogous argument as for the cube roots as in Lemma 3.2, while the condition (3.5a) follows from the \(4\times 4\) Frobenius-Stickelberger detereminant (A.12), setting the arguments of the elliptic functions equal to the four elliptic quartic roots of unity.
Similarly, proceeding along the same line, we can, from the higher order identity (C.7) of Appendix C, derive the equation for the elliptic quintic root of unity. Thus, setting
together with \(\delta +\omega _1(\delta )+\omega _2(\delta )+\omega _3(\delta )+\omega _4(\delta )\equiv 0(\text {mod period lattice})\), we derive from (C.7)
where use has been made of the addition formula (A.10a), the relation (A.9), which implies also
and the well-known relation \(\wp '''=12\wp \wp '\).
Extrapolating from the latter cases of \(N=4\) and \(N=5\) cases, the general relation for the elliptic \(N{\textrm{th}}\) root of unity can be conjectured, but the proof requires increasingly complicated elliptic identities along the lines given in Appendix C. Thus, we can define:
Definition 3.3
The elliptic \(N{\textrm{th}}\) roots of unity are, up to the periodicity of the period lattice, defined as the roots of the following equation
with \(\omega _0(\delta ):=\delta \), and subject to the condition that they are independent of \(\kappa \).
That these elliptic roots \(\omega _i(\delta )\), (\(i=0,\dots ,N-1\)) exist follows from the fact that the function \(\wp ^{(N-2)}\) is an elliptic function of order N, and hence that the equation \(\wp ^{(N-2)}(\omega )-\wp ^{(N-2)}(\delta )\) has N roots (modulo the period lattice). Furthermore, from the Frobenius-Stickelberger determinant formula (A.12) of rank N it follows, as before, that we have
and from the higher order elliptic identities, and by inserting \(\kappa =-\omega _j\), \(j=0,1,\dots ,N-1\) we also find that \(\wp ^{(N-2)}(\delta )=\wp ^{(N-2)}(\omega _j)\ , \quad j=0,1,\dots N-1\ . \) The latter, can be taken as an alternative definition of the elliptic \(N{\textrm{th}}\) root of unity. Then from elliptic identities such as the ones presented in Appendix C, it can be shown that the following relations hold:
where \(\zeta ^{(l)}(x)=d^l\zeta (x)/dx^l\). We do not present here a formal proof of the above assertions, but merely indicate how this can be proven. In fact, the general form of identities of the type (C.1) and (C.7) which we need for the \(N{\textrm{th}}\) root of unity can be described as follows:
where the function \({\mathcal {F}}\) takes exactly the form of the expansion of the \((N-1){\textrm{th}}\) derivative of the Weierstrass \(\sigma \)-function:
when expanded in \(\zeta \), \(\wp \) and \(\wp '\). The expression \({\mathcal {F}}\) corresponds to this expansion, where whenever we have an odd function in this expansion (like \(\zeta \), \(\wp '\), etc.) we substitute the combination
(and similar for \(\wp '\)), and whenever we encounter \(\wp \) (we have to replace everywhere the higher derivatives of \(\wp \) by their expressions in terms of \(\wp \) and \(\wp '\) exclusively) we substitute the combination:
Subsequently, by substituting \(\kappa _j=\omega _j(\delta )\) and \(x=\kappa \) we obtain the identity (3.8) subject to (3.9). Conversely, we can revert the argument as in the above specific cases, and show that in return this is a sufficient condition for the set of relations (3.10) to hold.
4 Reductions from the KP System to the KdV and BSQ Systems
The fundamental KP relations can be summarised as follows. As a consequence of (2.3) the plane wave factors \(\rho _\kappa (n,m,l)\) and \(\sigma _{\kappa '}(n,m,l)\) are of the form
and we set as in (2.27) the quantity \(\xi =\xi (n,m,l)\) in the non-autonomous form \(\xi =\xi _0-n\delta -m\varepsilon -l\nu \), then the fundamental KP system from the relations in Sect. 2.1 takes the form:
with (2.26). (Note that here the shift on the object \(\widetilde{{{\varvec{U}}}_\xi }\) also implies a shift the variable \(\xi \), i.e., \(\widetilde{{{\varvec{U}}}_\xi }:=\widetilde{{{\varvec{U}}}}_{\widetilde{\xi }}\), and \(\widetilde{{{\varvec{u}}}_\kappa (\xi )}=\widetilde{{{\varvec{u}}}_\kappa }(\widetilde{\xi })\), and similarly for the other quantities depending on \(\xi \), and for the other lattice shifts.) Similar relations to (4.2) hold for the other shifts on the multidimensional lattice, replacing \(\widetilde{~}\) by \(\widehat{~}\) or \(\overline{~}\) and \(\delta \) by \(\varepsilon \) or \(\nu \) respectively.
Applying multiple shifts we get higher order relations, e.g. the second order relation
where, according to (C.1), we have the coefficients
and taking into account the quantity (A.9)
Note that the relations (4.3) do not depend on the intermediary steps (i.e., on single-shifted objects).
Remark
In general the following functions appear in the coefficients of the fundamental relations:
and they can be expressed in terms of \(\zeta \)-functions using the relations of Appendix C.
Similarly the third-order shift relations read:
where
and
(which remarkably does not depend on the intermediate values \(\widehat{\overline{\xi }}\) or \(\overline{\xi }\)). Note that the latter expression does not seem manifestly symmetric w.r.t. permutation of \(\delta \), \(\varepsilon \), \(\nu \), but nevertheless it is, as can be shown using (A.8) from the Appendix C. Furthermore, we recall
which can be expressed in terms of \(\zeta \) functions using the identities in Appendix C.
Finally, the relevant basic relations for the \(\tau \)-function read:
as derived in Appendix B.
We can now use these higher order shift relations to obtain the higher order dimensional reductions of the lattice KP system, by imposing periodicity w.r.t. the combined shifts where the shift parameters are related through the elliptic roots of unity. In this way we get for \(N=2\) (elliptic square root of unity) the reduction to the lattice KdV system, while for \(N=3\) (elliptic cube root of unity) we attain the reduction to the lattice BSQ system, and hence to the elliptic solutions of the latter.
4.1 Reduction to the lattice KdV system
The reduction from KP to KdV type systems is imposed by requiring that the following condition holds on the plane wave factors:
A remark is in order here as to the meaning of this relation. In fact, in the KP system each lattice parameter, say \(\delta \), can be associated with a direction in a lattice of arbitrary dimension. However, when acting on KP solutions in general position, the shifts \(T_\delta \) and \(T_{-\delta }\) are associated with two different directions in that lattice, \(\delta \) with a lattice variables \(n=:n_\delta \) and \(-\delta \) with another discrete variable \(n_{-\delta }\) independent of \(n_\delta \). However, identifying the variable \(n_{-\delta }\) with the reverse direction of the variable \(n_\delta \) through the relation (4.6) we force a dimensional reduction on the system, which essentially reduces the 3-dimensional KP lattice system to a 2-dimensional system which is the lattice KdV system. Obviously we will still have the KP equation valid between any three shifts, but by using (4.6) any KP type equation between shifts in the \(n_\delta \) and \(n_{-\delta }\) directions will trivialise.
Note, furthermore, that in (4.6) we have effectively employed the elliptic square roots of unity \(\omega _0(\delta )=\delta \) and \(\omega _1(\delta )=-\delta \) as explained in Sect. 3, and (4.6) amounts to the assertion that the reverse operation to any shift in the multidimensional lattice generated by shifts \(T_\delta \) (for all \(\delta \)) corresponding to a parameter \(\delta \) acts in the same way on the dependent variables as the shift corresponding to the parameter \(-\delta \). Hence in the KP integral equations (2.2) we need to restrict ourselves to integration over suitable measure and contour, say \(d\mu (\ell )\) and \(\varGamma \), where this identification is implemented. Thus, the integral equation for the KdV class, reads:
with the corresponding expression for the matrix \({{\varvec{U}}}_\xi \):
Thus, the reduction to the lattice KdV system is obtained by implementing the relations
which follow as additional conditions on the fundamental system of KP equations given in Sect. 2. The relations for the KdV class can be obtained in this way, and comprise:
where the first two relations are exactly the same as in the KP case, but where the last additional relation is obtained from the condition (4.9). We note that the reduction to the KdV case can also be interpreted as the condition that the infinite matrix \({{\varvec{U}}}_\xi \) is symmetric under transposition, i.e. \(({{\varvec{U}}}_\xi ) ^T={{\varvec{U}}}_\xi \) . Furthermore, in this reduction we have purely algebraic relations (i.e., without involving any lattice shifts), which are obtained from the double-shift relations (4.3) by taking \(\varepsilon =\omega (\delta )\) and taking into account the condition (4.9). Thus, we get
The resulting equations in this case are the equations in the KdV class of lattice equations, namely two-dimensional lattice equations involving only two lattice directions. Thus, as an example, we mention here only the lattice (potential) KdV equation, cf. [30], in the form
In [26] it was shown how to obtain, from an elliptic Cauchy matrix scheme, elliptic multi-soliton solutions for all ABS equations [2], apart from Q4 (solitons of the latter were obtained by a different approach in [5]). In principle the DL scheme covers an even wider class of solutions. Since in this paper we concentrate on the higher-rank elliptic reductions, we will refrain here from deriving all the lattice equations from the above scheme and refer to [26] for details.
4.2 Reduction to the lattice BSQ system
The dimensional reduction to the BSQ case (i.e., the case \(N=3\)) from the lattice KP case is obtained by imposing the condition
which leads to the identifications \(\kappa '=-\omega _j(\kappa )\) (\(j=0,1,2\)) in the integral equation (2.2), where we have set for convenience \(\omega _0(\kappa ):=\kappa \). A similar remark as in the KdV reduction case applies to the relation (4.13), namely that for generic solutions of the KP system the lattice directions associated with lattice parameters \(\delta \), \(\omega _1(\delta )\) and \(\omega _2(\delta )\) should be viewed as distinct, and act on three independent lattice variables \(n_\delta \), \(n_{\omega _1(\delta )}\) and \(n_{\omega _2(\delta )}\) respectively. However, in special solutions of the KP system, on which the three-fold condition (4.13) holds, there is an implicit identification between these lattice variables and their corresponding elementary shifts.
It is here where the elliptic cube root of unity is seen to play the key role. In fact, from the defining relation (3.2) we have
and similarly, we have that
Hence, we get the condition (4.13) to hold provided that \(\kappa '=-\omega _j(\kappa )\) (\(j=0,1,2\)). The latter is the condition on the spectral variable which reduces the integral equations (2.2) to the ones for the BSQ case.
Thus, Eq. (4.13) when implemented on the KP integral equations (2.2), e.g. by restricting the measure \(d\mu (\ell ,\ell ')\) to contain \(\delta \)-functions of the form \(\delta (\ell '+\omega _j(\ell ))\) (\(j=1,2\)), leads to the following integral equation for the BSQ elliptic solutions:
where \(\xi \) is given as in (2.27)—which guarantees that
and where \(\varGamma _j\) and \(d\mu _j(\ell )\) are contours and measures that need to be suitably chosen (see Sect. 6 for particular cases of choices for \(\varGamma _j\) and \(d\mu _j(\ell )\)). Furthermore, as a consequence of the reduction the eigenvector \({\varvec{u}}_\kappa \) obeys
Together with the formal integral equation (4.14) we have in this reduced case the following formula for the infinite matrix \({{\varvec{U}}}_\xi \):
As a consequence of the analysis of the KP system, which fully goes through modulo the reduction, we can now readily inherit all the fundamental relations for the infinite-matrix scheme for the BSQ system by implementing the constraint (4.15) on the KP system (4.2) in the multidimensional space of lattice variables. By assuming that this multidimensional grid includes the lattice directions associated with the parameters \(\omega _1(\delta )\) and \(\omega _2(\delta )\), subject to the condition (3.2), we get the following fundamental infinite matrix system for the elliptic BSQ solution. First, the discrete dynamics for any direction indicated by the parameter \(\delta \), is given by the relations:
for the forward shift (which is the same relation as for the KP system), while for the reverse shift we have additional relations:
and similar relations for the other lattice directions. These follow directly from the double-shift KP relations (4.3) implementing the triple shift conditions on the reduction (4.15). Second, the condition (4.15) gives rise to the following algebraic constraints (which involve no lattice shifts):
which follow from (4.4) implementing the condition (4.15), and noting that
The system of equations (4.17) in all relevant lattice directions form the fundamental system from which the BSQ lattice system, including the corresponding Lax pairs, can be derived, thus implying that the quantities we will extract from the infinite matrix \({{\varvec{U}}}_\xi \) of (4.16) the elliptic class of solutions of those lattice systems.
The next step, in order to derive from the set of relations (4.17) closed-form equations in terms of single elements of the matrix \({{\varvec{U}}}_\xi \) or combinations thereof, is to identify the relevant entries and to specify the relations involving those entries. Then closed-form equations are obtained by combining those relations for two different lattice shifts associated with two different lattice directions. For the sake of transparency of the structure, we will introduce yet another set of notations, which will make the relations as close as possible to the ones in the rational case. Thus, we introduce the notations:
i.e. \(\xi \)-dependent raising operators acting from the left and right, and
\(\xi \)-dependent lattice ‘parameters’ (replacing in a sense the lattice parameters p and q of [29]). In terms of these the fundamental relations (4.17) can be rewritten as
(and a similar set of relations with \(p_\xi \rightarrow q_\xi \) and \(\widetilde{~}\rightarrow \widehat{~}\)), with
and similarly for the other lattice directions. Some useful properties of these quantities are given by
and similarly for the quantities \(q_\xi \) with \({\varvec{\varLambda }}_{\widetilde{\xi }}\) replaced by \({\varvec{\varLambda }}_{\widehat{\xi }}\) and \(\delta \) by \(\varepsilon \).
5 Lattice BSQ Systems
In the previous section we have obtained the set of fundamental shift relations, in particular (4.19), in terms of one particular lattice shift. However, these relations are still not in closed form, since they involve the elliptic raising operators \({\varvec{\varLambda }}_\xi \) and \(\,^{t\!}{\varvec{\varLambda }}_\xi \). In this section we will endeavour to eliminate these operators by combining the relations for different shifts on a two-dimensional lattice associated with two lattice parameters \(\delta \) and \(\varepsilon \), and thus obtain some concrete closed-form lattice equations within the BSQ class, from the fundamental relations in terms of the infinite matrix \({{\varvec{U}}}_\xi \). The corresponding Lax pairs for those equations are constructed in Appendix D.
5.1 Derivation of the regular lattice BSQ equation
We proceed by constructing a system of closed-form equations for specific elements of the matrix \({{\varvec{U}}}_\xi \) by combining the relations from the system (4.17) for two different lattice shifts with parameters \(\delta \) and \(\varepsilon \). Setting by definition:
we can derive various relations from the system (4.19) by taking the \((~)_{0,0}\) element, either directly or after multiplication by factors of the form \(p_\xi +\,^{t\!}{\varvec{\varLambda }}_{\widetilde{\xi }}\) (from the right) or \(p_\xi -{\varvec{\varLambda }}_\xi \) (from the left). Thus, from (4.19a) and (4.17c) as well as (4.19c) we obtain the following set of relations (suppressing the arguments \(\xi \) of the functions):
Similarly from their counterparts for the other shift, i.e. replacing the shift \(\widetilde{~}\) by the shift \(\widehat{~}\) and \(p_\xi \) by \(q_\xi \), we obtain:
There exist several consistencies among these relations. For instance, subtracting (5.3b) from (5.2b) and using (5.2c) and (5.3c) to eliminate \(u_{1,1}\) we find a triviality provided the following relation
which follows from the definition of these quantities, is being used. Another relation that is frequently needed in these computations is
which is a consequence of
which in turn follows from the general relation (A.14) of Appendix A.
Combining the above relations in a nontrivial way, while eliminating \(u_{0,2}\) and \(u_{2,0}\) and \(u_{1,1}\), is obtained by subtracting (5.3c) from (5.2c) and using the \(\widetilde{~}\) -shift of (5.3d) and the \(\widehat{~}\) -shift of (5.2d), which leads to the following relation:
which together with (5.2a) and (5.3a) form a coupled 3-component system for the functions \(u_{0,0},u_{1,0}\) and \(u_{0,1}\) which is equivalent to the lattice BSQ system. In fact, by eliminating the quantities \(u_{1,0}\) and \(u_{0,1}\) in this three-component lattice system, by shifting (5.5) in both lattice directions, we obtain the 9-point equation:
In Sect. 6, we will relate (5.6) to the ‘normalised’ form of the lattice BSQ equation (1.2) and show that the DL scheme leads to explicit solutions, namely of elliptic seed and soliton type.
5.2 Parameter-dependent quantities and the lattice modified BSQ equation
In order to obtain closed-form equations from the shift relations (4.17a) and (4.17c), we introduce the following objects:
and recall \(v_\alpha (\xi ), w_\beta (\xi ), s_{\alpha ,\beta }(\xi )\) defined in Sect. 2.3 :
Several identities are needed to derive the relevant single-shift relations for these quantities, namely
which follow from (A.13), as well as
which follow from (A.10a) and (A.8) in combination with (5.8). Using (5.8) by multiplying (4.17a) by factors \(\left( \chi ^{(1)}_{\alpha ,\widetilde{\xi }}({\varvec{\varLambda }})\right) ^{-1}\) from the left and by \(\left( \chi ^{(1)}_{\beta ,\xi }(\,^{t\!}{\varvec{\varLambda }})\right) ^{-1}\) from the right and taking the \((~)_{0,0}\) element, and using the relations (5.9) by multiplying (4.17c) by factors \(\left( \chi ^{(1)}_{\alpha ,\xi }({\varvec{\varLambda }})\right) ^{-1}\) from the left and by \(\left( \chi ^{(1)}_{\beta ,\widetilde{\xi }}(\,^{t\!}{\varvec{\varLambda }})\right) ^{-1}\) from the right and taking the \((~)_{0,0}\) element, we obtain the following relations relating the 2-parameter quantity \(s_{\alpha ,\beta }(\xi )\) to the other quantities defined above:
respectively, in which, for convenience, we have introduced the notations:
The latter quantities appear in (5.10b) in the following form:
(noting the curious interchange of the roles of \(\xi \) and \(\widetilde{\xi }\) when \(\delta \) changes into \(-\delta \)). Note that we also have the relations:
For the single-parameter quantities \(s_\alpha , t_\beta ,r_\alpha ,z_\beta \) we can derive a system of shift relations in a similar way, using single-parameter multiplying factors from the left or the right before taking the \((~)_{0,0}\) elements. Thus, from (4.17a) multiplying from the left by \(\left( \chi ^{(1)}_{\alpha ,\widetilde{\xi }}({\varvec{\varLambda }})\right) ^{-1}\) or by \(\left( \chi ^{(1)}_{\beta ,\xi }(\,^{t\!}{\varvec{\varLambda }})\right) ^{-1}\) from the right and projecting on the \((~)_{0,0}\) element, we obtain:
Furthermore, by multiplying (4.17a) from the left by \(\left( \chi ^{(1)}_{\alpha ,\widetilde{\xi }}({\varvec{\varLambda }})\right) ^{-1}\) and from the right by \(\chi ^{(1)}_{\delta ,\widetilde{\xi }}(\,^{t\!}{\varvec{\varLambda }})\) and taking the \((~)_{0,0}\) element, respectively by multiplying from the right by \(\left( \chi ^{(1)}_{\beta ,\xi }(\,^{t\!}{\varvec{\varLambda }})\right) ^{-1}\) and from the left by \(\chi ^{(1)}_{-\delta ,\xi }({\varvec{\varLambda }})\) and projecting, we obtain
Finally, from (4.17c) by multiplying from the left by \(\left( \chi ^{(1)}_{\alpha ,\xi }({\varvec{\varLambda }})\right) ^{-1}\), or from the right by \(\left( \chi ^{(1)}_{\beta ,\widetilde{\xi }}(\,^{t\!}{\varvec{\varLambda }})\right) ^{-1}\) and taking the \((~)_{0,0}\) elements, while using the relations (5.9a) and (5.9b) respectively, we obtain
Henceforth we will, for brevity, suppress the arguments \(\xi \) in these parameter functions, and simply write \(v_\alpha \) for \(v_\alpha (\xi )\), etc., where it is understood that shifts \(\widetilde{v_\alpha }=\widetilde{v_\alpha (\xi )}\) act on the function as well as its argument, unless explicitly stated.
Obviously the relations (5.14)–(5.16) have their counterpart for the other lattice direction, replacing \(\widetilde{~}\) by \(\widehat{~}\) and \(\delta \) by \(\varepsilon \). By combining relations of this type for the two lattice directions, and thereby eliminating some of the parameter quantities, various significant additional relations of Miura type can be derived. Here the relations (5.2a) and (5.3a) are needed as well to eliminate the quantities \(u_{0,0}\), \(u_{1,0}\) and \(u_{0,1}\) when necessary. For instance, from (5.16) by eliminating the quantities \(r_\alpha \) and \(z_\beta \) and using (5.2a) and (5.3a) we find the 2-shift relations
where we have also used (5.4a) and the relation
which follows from the definitions of these quantities and the relation (A.8). At the same time, from (5.14) and its counterparts with shifts in the other lattice direction, we can derive the relations:
as well as the relations
We note at this point that we have now already two closed systems of equations, namely one for the quantities \(u_{0,0}\), \(v_\alpha \) and \(s_\alpha \) given by (5.17a), and the two relations given by (5.14a) and its counterpart in the other lattice shift direction, and the other for \(u_{0,0}\), \(w_\beta \) and \(t_\beta \) (5.17b), and the two relations given by (5.14b) with its counterpart in the other lattice direction. However, one can also derive at this juncture a closed-form 9-point equation for \(v_\alpha \), or alternatively for \(w_\beta \), alone. In fact, by solving \(s_\alpha /v_\alpha \) from (5.17a), and inserting the result into (5.14a), we get
and by subtracting a second copy of this relation with \(\delta \) and \(\varepsilon \) interchanged (interchanging also \(p_\xi \) and \(q_\xi \) and the \(\widetilde{~}\) and \(\widehat{~}\) lattice shifts) we can use (5.19a) to eliminate all the quantities \(u_{0,0}\). The result is a 9-point equation for \(v_\alpha \), namely
which can be brought into the form of the lattice modified BSQ equation of [29].
Similarly, solving \(\widehat{\widetilde{t_\beta /w_\beta }}\) from (5.17b) and inserting this into (5.14b) we get
and subtracting the shifted versions of this relation in both lattice directions, we can eliminate the quantities \(u_{0,0}\) by using (5.19b) to get a closed-form 9-point relation for \(w_\beta \). The result is:
In Sect. 6 we shall bring this equation in the standard form of lattice modified BSQ and present some explicit elliptic type solutions.
5.3 Trilinear equation of the \(\tau \)-function for the lattice BSQ system
The \(\tau \)-function associated with the lattice BSQ system is exactly the same as given for the lattice KP system, introduced in Sect. 2.2, namely given by (2.10), but of course taking into account the BSQ reduction discussed in Sect. 4.2. The principal relations this \(\tau \)-function obeys were derived in Appendix B, and are given by te shift relations
where the overtilde \(\widetilde{\tau _\xi }\) and the undertilde 
denotes the forward and backward shift \(T_\delta \) and \(T_\delta ^{-1}\) respectively, implemented also on the argument \(\xi \), and similarly for the overhat \(\widehat{\tau _\xi }\) and underhat 
, associated with the forward and backward shifts \(T_\varepsilon \) and \(T_\varepsilon ^{-1}\) respectively, acting also on the argument \(\xi \). As a consequence of (5.24), together with the relations
(which are equivalent to (5.10)), setting \(\alpha =\delta , \beta =-\varepsilon \), we also have
Furthermore, from (5.19b), setting \(\beta =-\delta \), together with (5.24), we get
while from (5.14a) setting \(\alpha =\delta \), which implies \(a_{\widetilde{\xi }}=-p_\xi \), we see that
The latter inserted into (5.17a), with \(\alpha =\delta \), implying \(A_\delta (\widetilde{\xi })=P_\delta (\xi ),~A_\varepsilon (\widehat{\xi })=P_\varepsilon (\widehat{\xi })\), gives us:
where now the coefficients \(A_\delta (\widetilde{\xi })\) and \(A_\varepsilon (\widehat{\xi })\) when \(\alpha =\delta \) are given by
Adding to the latter relation the one in which the \(\widetilde{~}\) and \(\widehat{~}\) shifts and the \(\delta \) and \(\varepsilon \) parameters are interchanged, thereby introducing the coefficients
and eliminating the \(u_{0,0}\) using the relation (5.27), we arrive at a trilinear equation for the \(\tau \)-function, namely
Taking \(\tau _{\xi }=\frac{f}{\sigma (\xi )}\) in (5.28), we derive an autonomous trilinear equation
A similar trilinear equation like (5.29) was found in [40] for the extended lattice BSQ system in the case of the rational parametrisation of the equation.
5.4 2-parameter variables and generalised lattice Schwarzian BSQ equation
Now we will focus on the relations (5.10) for the quantity \(s_{\alpha ,\beta }(\xi )\) and derive a closed-form equation for that quantity.
First, we want to rewrite the relations in a form that is more suitable for a comparison with the standard ABS class of quad-lattice equations, cf. [2]. To achieve this we introduce the quantitiesFootnote 5
The corresponding shift relations following from (5.10), or equivalently (5.10b), read
(we omit the arguments \(\xi \) in the main functions, while keeping them in the coefficients, where it is understood that all the shifts acting on those functions also act on that argument \(\xi \) of the functions), and similarly the corresponding counterparts in the other lattice direction, namely
Next, taking the \(\widehat{~}\) shift of (5.31b) and using (5.17b) and the \(\{\widehat{~}, \varepsilon \}\) version of (5.14a) to replace \(\widehat{\widetilde{t_\beta (\xi )}}\) and \(\widehat{s_\alpha (\xi )}\), respectively, and using (5.19b) to replace \(p-q+\widehat{u_{0,0}}-\widetilde{u_{0,0}}\), we obtain
Multiplying numerator and denominator in the fraction on the right-hand side by \(\widehat{\widetilde{v_\alpha }}\) we can use (5.32a) to express the entire right-hand side in terms of \(S_{\alpha ,\beta }\), which leads to
where in the denominator some of the terms have disappeared by virtue of the relation
and where the quad function in the numerator is given by
where we have used (5.13) and where the coefficients \(\gamma \), \(\gamma '\) and \(\gamma ''\) are given by
where we have used the identitiesFootnote 6
(and similarly with a’s replaced by b’s). The latter are a consequence of the relations
and similar relations with a and b and \(\delta \) and \(\varepsilon \) interchanged.
We can bring the quadrilateral \({\mathcal {Q}}\) into a more standard form by performing a further (gauge) transformation, namely
where the function \(\phi _{\alpha ,\beta }(\xi )\) solves the compatible system of first order ordinary difference equations
Implementing the change of variables (5.37) the relation (5.34) changes into
where now
in which
Furthermore, rewriting (5.31a) in terms of \({\mathfrak {u}}_{\alpha ,\beta }\) we have
Equation (5.38), together with (5.40) and its counterpart in the other direction (with \(\delta \) replaced by \(\varepsilon \) and \(\widetilde{~}\) replaced by \(\widehat{~}\) ) forms a closed quadrilateral system for the three variables \({\mathfrak {u}}_{\alpha ,\beta }\), \(v_\alpha \) and \(w_\beta \), which is the most general form of the lattice BSQ system. However, by using the identity
and inserting the expressions for the quantities in brackets in terms of \({\mathfrak {u}}_{\alpha ,\beta }\) one can write down a closed form 9-point equation in terms of \({\mathfrak {u}}_{\alpha ,\beta }\) alone, see [40]. In the elliptic case that equation takes exactly the same form, except for the parametrisation of the lattice parameters involved, namely
The latter equation, which can be thought of as a generalisation of the lattice Schwarzian BSQ equation, [24], contains both the ‘regular’ lattice BSQ equation, (1.2) as well as the lattice modified BSQ equation as special coalescence limits on the parameters \(\alpha \) and \(\beta \). Thus, the 9-point equation is the most general lattice BSQ system known so far.
6 Elliptic Seed and Soliton Solutions of the Lattice BSQ Systems
In the previous section we have obtained various lattice equations in the BSQ class, like the regular lattice BSQ equation (5.6), the lattice modified BSQ equations (5.21) or (5.22) and the generalised Schwarzian BSQ equation (5.41), as well as the trilinear equation for the \(\tau \)-function (5.28). From the DL structure we automatically have a large class of elliptic solutions for those equations, including soliton solutions and inverse scattering type solutions. However, the equations we obtained are not yet in ‘standard form’, cf. [12], in that they contain elliptic coefficients. In this section we will perform point transformations to bring the equations in the required standard form, and in doing so we get the elementary elliptic seed solutions of the standard lattice BSQ systems. Furthermore, we present some explicit one- and two-elliptic soliton solutions, and demonstrate the general structure of the elliptic multi-soliton solutions in terms of the corresponding Cauchy matrix scheme.
6.1 Seed and elliptic 1-soliton solutions of the lattice BSQ system
Consider the following point transformation
with
and \(u=u(n,m),~v=v(n,m),~w=w(n,m)\). Imposing this transformation on (5.2a), (5.3a) and (5.5), i.e.,
we readily obtain the following three-component lattice BSQ system
which was referred to as the (B-2) system in Ref. [13], but which appeared also in [35, 37]. Thus, we attain the following result on the 0-soliton solution (0SS) of the lattice equations.
Proposition 6.1
The system (6.4) admits the following elliptic ‘seed’ solution
where \(\xi =\xi _0-n\delta -m\varepsilon \) and \(\xi _0\) is an arbitrary initial value.
By eliminating v and w in the lattice BSQ system (6.4), we also get the lattice BSQ equation (1.2), rewritten as
together with its elliptic seed solution (6.5a).
Meanwhile, it is noted that (6.6) can be transformed into (5.6) by the transformation (6.1a) and (6.2a). Moreover, making use of the seed solution (6.5) and the Bäcklund transformations (see [34] and [40]), we can construct the elliptic 1-soliton solution (1SS) as follows.
Proposition 6.2
The system (6.4) admits the following elliptic 1-soliton solution (with soliton parameter \(\kappa \))
where \(\omega _{1}\) and \(\omega _{2}\) are the elliptic cube roots of unity (see Sect. 3) subject to \(\kappa +\omega _1(\kappa )+\omega _2(\kappa )\equiv 0(\text {mod period lattice})\) and where \(\rho _i~(i=1,2)\) are plane wave factors given by
In particular the 9-point equation (6.6) admits the solution given by (6.7a).
6.2 Elliptic seed solution for the lattice modified BSQ equation
To derive the 3-component form of the lattice modified BSQ equation as in [13], we consider the following point transformation
with new functions \(U=U(n,m),~V=V(n,m),~Y=Y(n,m)\), and where \(x_0\) is given by
and C is an arbitrary complex constant. By using this transformation, (5.14a) and its \(\{\widehat{~},\varepsilon \}\) counterparts and (5.17a) are turned to the following system corresponding to the (A-2) system of [13]
in which we have used the notation in (5.39). Eliminating U and Y in the above system, we obtain the lattice modified BSQ equation, written as the 9-point equation, [29],
Alternatively, for the system (5.17b) and its \(\{\widehat{~},\varepsilon \}\) counterparts and (5.14b), we employ the following transformation
and obtain
with \(U=U(n,m),~W=W(n,m)\) and \(Z=Z(n,m)\) functions of n, m and C a constant, whereas \(x_0 \) was defined in (6.10) appearing before in (6.9). Eliminating U and Z in (6.14) or noticing the reversal symmetry (see [13]) of the system (6.11) we have an alternate form of the lattice modified BSQ equation, written as
which is related to the equation (5.22) through the transformation (6.13b).
The seed solutions (i.e., the 0-soliton solutions of the lattice equations) can be read off from the change of dependent variables (6.9) and (6.13), by removing \({{\varvec{U}}}_\xi \) in the definitions (5.7). Thus, we arrive at the following result concerning the seed solutions.
Proposition 6.3
The lattice modified BSQ equation (6.12) admits the following elliptic seed solution
Similarly, the alternative lattice modified BSQ (6.15) admits the elliptic seed solution
The elliptic 1-soliton solutions can be readily constructed as well from the DL framework, but we will omit the explicit formulae here, as we will present the general elliptic Cauchy matrix forms in Sect. 6.4 below.
6.3 Elliptic seed solution for the lattice Schwarzian BSQ equation
Now we want to present the lattice Schwarzian BSQ equation from the three-component lattice system, i.e, (5.31a), (5.32a) and (5.33). Employing the following point transformation to this system
where \(V=V(n,m),~W=W(n,m)\) and \(H=H(n,m)\) are functions of n, m, we get the following system corresponding to the (C-3) system in Ref. [13]
which leads to the lattice Schwarzian BSQ equation, written as
From the transformations (6.18c) and (5.30) and the definition of \(s_{\alpha ,\beta }(\xi )\) given by (5.7g), we deduce the following
Proposition 6.4
The lattice Schwarzian BSQ equation (6.20) admits the following elliptic seed solution
In the next subsection we will present the general elliptic Cauchy matrix framework from which the elliptic multi-soliton solutions can be readily found in a closed form.
6.4 Cauchy matrix scheme for elliptic multi-soliton solutions
We now consider the elliptic multi-soliton solutions, which we obtain by choosing a particular measure, namely in (4.14)
where \(\varLambda _{j,j'}\) are residues of the measures \(d\mu _j(\ell )\), and choosing the \(\varGamma _j~(j=1,2)\) to be contours in the complex plane surrounding the singularities \(\{\kappa _{j,j'}\}\). In this case the linear integral equation (4.14) is reduced to a matrix system of the form
Setting \(\kappa =\kappa _{i,i'}\), where \(i=1,2\), \(i'=1,\dots , N_{i}\) in Eq. (6.23), we then have a linear system for the quantities \(u_{\kappa _{i,i'}}\) from the equation (6.23), i.e.,
which can be rewritten as
where
\({{\varvec{I}}}_{{\mathcal {N}}}\) is an identity matrix of size \({\mathcal {N}}=N_1+N_2\) and \({\varvec{M}}\) is a 2\(\times \)2 block generalised Cauchy matrix with rectangular blocks of size \(N_j\times N_i~(j,i=1,2)\) with elements
We can now make the soliton solutions explicit by assuming that the coefficients \({\varvec{\varLambda }}_{j,j'}\) are chosen such that the matrix \({{\varvec{I}}}_{{\mathcal {N}}}+{\varvec{M}}\) is invertible, in which case solutions to (6.23) could be written out explicitly. Making use of (4.16), (6.22) and (6.25), we can also obtain the explicit expression for \({{\varvec{U}}}_\xi \) written as
with elements
in which
and \({\varvec{K}}\) and \({\varvec{L}}\) are the diagonal matrices given by
and \({\varvec{r}},{{\varvec{s}}}\) are row vectors given by
By using the relation (2.5) and the above expression for \({{\varvec{U}}}_\xi \), we can write the quantities introduced in (5.7) in the following way:
Note that in Eq. (6.27) the matrices \(\wp ({\varvec{K}})\), \(\eta _\xi ({\varvec{L}})\), \(\chi ^{(1)}_{\alpha ,\xi }({\varvec{K}})\) and \(\chi ^{(1)}_{\beta ,\xi }({\varvec{L}})\) denote the diagonal matrices with as entries the corresponding elliptic functions evaluated at the entries of the corresponding matrices \({\varvec{K}}\) or \({\varvec{L}}\), e.g.
etc. Note that all these expressions (6.27) can be written out explicitly after choosing the various parameters of the solution. In this sense, \(\tau \)-function can also be expressed by
where the explicit expression can be computed by means of the usual expansion methods for determinants.
7 Conclusions
In this paper we constructed a rich class of elliptic solutions of a family of BSQ type partial difference equations, in particular the lattice BSQ equation (1.2), the lattice modified BSQ equations (5.21) and (5.22), and the lattice Schwarzian BSQ equation (6.20), while the most general equation is the generalised lattice Schwarzian BSQ equation (5.41), and the underlying equation for the \(\tau \)-function is given by trilinear equation (5.28). The various equations, which in their scalar form are difference equations on a 9-point stencil, can also be written in their multi-component forms, (see Sect. 6). The direct linearisation (DL) framework, equipped with an elliptic Cauchy kernel, produces in particular elliptic multi-soliton solutions (when the integration in the integral equations such as (4.14) together with (4.16) only involve the residues at simple poles in the spectral variable, see Sect. 6.4), but more generally, for more general integration measures \(d\mu _j(\ell )\) and integration contours \(\varGamma _j\) in (4.14) it comprises also inverse scattering type solutions.
A salient feature of the DL approach is that it incorporates several integrable models within one framework through the infinite matrix (4.16), the distinct entries of which obey various Miura related equations. The structure of the BSQ system (or more generally of the \(N{\textrm{th}}\) GD hierarchy) was obtained performing a dimensional reduction on the KP system, by imposing a constraint of the type (4.13). However, where in the case of rational Cauchy kernel this would involve the emergence of cube roots of unity, in the elliptic case one needs an elliptic analogue of the notion of cube root of unity for the reduction to work. These, or more generally the elliptic \(N{\textrm{th}}\) root of unity, were defined in Sect. 3. Unlike the conventional roots of unity, these elliptic analogues depend intrinsically on a parameter - in the case at hand, the lattice parameter. This novel concept of elliptic \(N{\textrm{th}}\) root of unity may well have a significance in other areas of mathematics, e.g. cyclotomic fields.
Whereas we restricted ourselves in this paper to solutions of the BSQ class of lattice systems, the main underlying structural relations were given for the entire lattice GD hierarchy, and thus the results obtained may be readily generalised to the higher order systems in this hierarchy, although the development of the explicit formulae and corresponding computations may fast become quite overwhelming. The present paper has made clear that the KdV case (\(N=2\)) of the GD hierarchy is not really representative of what goes on in the general case, while the BSQ case (\(N=3\)) provides a first good insight into the real intricacies of the general construction. Thus, in spite of the importance of the famous ABS classification results of [2], the present work demonstrates that there is ‘integrable life beyond ABS’ and that in many respects the results on the latter systems do not really provide the necessary insights for multicomponent and higher-order integrable lattice systems.
A particular generalisation that we have not considered (yet) in the present paper, is that of the so-called extended BSQ systems. These involve parameter-generalisations of the lattice BSQ systems of [29] found by Hietarinta in [13]. It was shown in [40], in the rational case, how these parameter generalisations fit into the DL framework, thereby providing a rich class of solutions of those extended BSQ systems, cf. also [14] for a recent review. In fact, those extensions arise from an unfolding of the dispersion curve from \(G(k,\omega )=\omega ^3-k^3=0\) to \(G(k,\omega )=k^3-\omega ^3+\alpha _2(\omega ^2-k^2)+\alpha _1(\omega -k)=0\) where the parameters \(\alpha _1\) and \(\alpha _2\) are the extension parameters, the inclusion of which effectively embeds the (\(N=2\)) KdV system into the (\(N=3\)) BSQ system. In the elliptic case, this extension would require an unfolding of the elliptic cube root of unity condition, to a more general condition on the roots \(\omega _j(\delta )\), (\(j=0,1,\cdots ,N-1\)), namely
(possibly with \(\alpha _{N-2}=1\)) but we leave the development of the relevant explicit formulae involving the extension parameters \(\alpha _j\) corresponding to a future publication.
As a final remark we need to point out that in this paper we have considered elliptic solutions of equations that in themselves are rational expressions of the dependent variables and their shifts. There are, however, also equations that one could characterise as elliptic integrable systems, i.e. equations which are either parametrised through elliptic functions (i.e. where the dependent variable enters in the argument of elliptic functions) or where the equation, albeit of rational form, contains an algebraic curve in terms of the dependent variable. Prime examples of the latter are the Krichever–Novikov equation or the fully anisotropic Landau–Lifschitz equations, cf. e.g. [6, 19, 20, 22, 32] and their discrete analogues [1, 8, 28]. It is not clear at this stage how these two matters, elliptic equations versus elliptic solutions, are connected, but some hint as to this connection may be in the elliptic versions of integrable many-body systems [3, 17, 18, 21], which can be viewed both as a class of elliptic solutions as well as elliptic integrable systems in their own right.
Notes
We note that in [8] a higher rank elliptic system was proposed generalising the 3-leg form of Q4, but it is not clear yet what is the rational form of that system.
For clarity of notation we indicate the spectral parameter dependence by a suffix rather than as an argument of a function.
To make sense of the infinite determinant in (2.10) we can use the expansion formula
$$\begin{aligned} \det ({{\varvec{1}}}+A)=1+\sum _i A_{ii} + \sum _{i<j} \left| \begin{array}{cc}A_{ii}&{}A_{ij}\\ A_{ji}&{}A_{jj}\end{array}\right| + \dots \ , \end{aligned}$$which is valid if \(A={\varvec{\varOmega }}_\xi \cdot {{\varvec{C}}}\) is of finite rank (which is the case for the solutions given in Sect. 6). In general the requirement of these factors to be of finite rank imposes some conditions on the integrations in (2.6), in view of the fact that all terms in the expansion are of the form \( \textrm{tr}_{{\mathbb {Z}}}\left( ({\varvec{\varOmega }}_\xi \cdot {{\varvec{C}}})^n\right) \) for some integer power n, and where \(\textrm{tr}_{{\mathbb {Z}}}\) means and infinite trace over all integer-labeled diagonal entries.
In [36] these quantities were already mentioned (in Remark 7, on p. 1037 of that paper), but without pushing the notion of ‘elliptic cube roots’. Here we argue that, more generally, a notion of ‘elliptic \(N{\textrm{th}}\) root of unity’ makes perfect sense.
From (5.26) we see that the quantity \(S_{\alpha ,\beta }\) could readily expressed in the \(\tau \)-function if we allow there to lattice shifts associated with the parameters \(\alpha \) and \(\beta \) in lieu of \(\delta \) and \(\varepsilon \). This would require, by MDC, that we have to extend the plane wave factors \(\rho _\kappa \) and \(\sigma _{\kappa '}\) in the DL scheme to contain factors with \(\alpha \) and \(\beta \) as lattice parameters, and hence directions in the lattice with discrete variables associated with these additional lattice directions.
These identities are also consequences of the chain of relations
$$\begin{aligned} \frac{\chi ^{(1)}_{\alpha ,-\varepsilon }(\widetilde{\xi }) }{\chi ^{(1)}_{\alpha ,-\varepsilon }( \xi ) } =\frac{\chi ^{(1)}_{\alpha ,\delta }(\widetilde{\xi }) }{\chi ^{(1)}_{\alpha ,\delta }(\widehat{\widetilde{\xi }) } } = \frac{\chi ^{(1)}_{\alpha ,-\delta }(\widehat{\xi }) }{\chi ^{(1)}_{\alpha ,-\delta }( \xi ) }=\frac{\chi ^{(1)}_{\alpha ,\varepsilon }(\widehat{\xi }) }{\chi ^{(1)}_{\alpha ,\varepsilon }(\widehat{\widetilde{\xi }) } }, \end{aligned}$$which in turn follow from the expressions for the \(\chi ^{(1)}\) quantities in terms of the quantities \(\varPhi \) as in (2.26), and using the identifications
$$\begin{aligned} \chi ^{(1)}_{\delta ,\alpha }(\widetilde{\xi })= p_\xi -a_\xi \ , \quad \chi ^{(1)}_{-\delta ,\alpha }(\xi )= -(p_\xi +a_{\widetilde{\xi }})\ , \quad \chi ^{(1)}_{\delta ,\beta }(\widetilde{\xi })= p_\xi -b_\xi \ , \quad \chi ^{(1)}_{-\delta ,\beta }(\xi )= -(p_\xi +b_{\widetilde{\xi }})\ ,\nonumber \\ \end{aligned}$$and similarly in the other direction
$$\begin{aligned} \chi ^{(1)}_{\varepsilon ,\alpha }(\widetilde{\xi })= q_\xi -a_\xi \ , \quad \chi ^{(1)}_{-\varepsilon ,\alpha }(\xi )= -(q_\xi +a_{\widehat{\xi }})\ , \quad \chi ^{(1)}_{\varepsilon ,\beta }(\widetilde{\xi })= q_\xi -b_\xi \ , \quad \chi ^{(1)}_{-\varepsilon ,\beta }(\xi )= -(q_\xi +b_{\widehat{\xi }})\ . \end{aligned}$$
References
Adler, V.E.: Bäcklund transformation for the Krichever–Novikov equation. Int. Math. Res. Not. 1, 1–4 (1998)
Adler, V.E., Bobenko, A.I., Suris, Yu.B.: Classification of integrable equations on quad-graphs, the consistency approach. Commun. Math. Phys. 233, 513–543 (2003)
Airault, H., McKean, H.P., Moser, J.: Rational and elliptic solutions of the Korteweg–de Vries equation and a related many-body problem. Commun. Pure Appl. Math. 30, 95–148 (1977)
Akhiezer, N.I.: Elements of the Theory of Elliptic Functions. Translation of Mathematical Monographs, vol. 79. AMS, Providence (1990)
Atkinson, J., Nijhoff, F.W.: A constructive approach to the soliton solutions of integrable quadrilateral lattice equations. Commun. Math. Phys. 299, 283–304 (2010)
Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Landau–Lifshitz equation: solitons, quasi-periodic solutions and infinite-dimensional Lie algebras. J. Phys. A: Math. Gen. 16, 221–236 (1983)
Delice, N., Nijhoff, F.W.: On elliptic Lax pairs and isomonodromic deformation systems for elliptic lattice equations. In: Konno, H., Sakai, H., Shiraishi, J., Suzuki, T., Yamada, Y. (eds.) Representation Theory, Special Functions and Painlevé Equations. Advanced Studies in Pure Mathematics, vol. 76, pp. 487–525 (2018)
Delice, N., Nijhoff, F.W., Yoo-Kong, S.: On elliptic Lax systems on the lattice and a compound theorem for hyperdeterminants. J. Phys. A: Math. Theor. 48, 035206 (2015)
Fokas, A.S., Ablowitz, M.J.: Linearization of the Korteweg–de Vries and Painlevé II equations. Phys. Rev. Lett. 47, 1096–1099 (1981)
Fokas, A.S., Ablowitz, M.J.: The inverse scattering transform for multidimensional \((2+1)\) problems. In: Wolf, K.B. (ed.) Nonlinear Phenomena. Proceedings Oaxtepec, Mexico 1982. Lecture Notes in Physics, vol. 189, pp. 137–183. Springer, Berlin (1983)
Fu, W.: Direct linearization approach to discrete integrable systems associated with \({\mathbb{Z} }_{ {\cal{N} }}\) graded Lax pairs. Proc. R. Soc. A 476, 20200036 (2020)
Hietarinta, J., Joshi, N., Nijhoff, F.W.: Discrete Systems and Integrability. Cambridge Texts in Applied Mathematics, vol. 54. Cambridge University Press, Cambridge (2016)
Hietarinta, J.: Boussinesq-like multi-component lattice equations and multi-dimensional consistency. J. Phys. A Math. Theor. 44, 165204 (2011)
Hietarinta, J., Zhang, D.J.: Discrete Boussinesq-type equations. In: Euler, N., Zhang, D.J. (eds.) Nonlinear Systems and their Remarkable Mathematical Structures, vol. 3, pp. 54–108. CRC Press, Boca Raton (2021)
Hietarinta, J., Zhang, D.J.: Multisoliton solutions to the lattice Boussinesq equation. J. Math. Phys. 51, 033505 (2010)
Konopelchenko, B.G.: Introduction to Multidimensional Integrable Equations. Springer, Heidelberg (1992)
Krichever, I.M.: Elliptic solutions of the Kadomtsev–Petviashvili equation and integrable systems of particles. Funct. Anal. Appl. 14, 282–290 (1980)
Krichever, I.M.: Elliptic analog of the Toda lattice. Int. Math. Res. Not. 2000(8), 383–412 (2000)
Krichever, I.M., Novikov, S.P.: Holomorphic fiberings and nonlinear equations. Sov. Math. Dokl. 20, 650–654 (1979)
Krichever, I.M., Novikov, S.P.: Holomorphic bundles over algebraic curves and nonlinear equations. Russ. Math. Surv. 35, 53–79 (1981)
Krichever, I.M., Lipan, O., Wiegmann, P., Zabrodin, A.: Quantum integrable systems and elliptic solutions of classical discrete nonlinear equations. Commun. Math. Phys. 188, 267–304 (1997)
Mikhailov, A.V.: The Landau–Lifschitz equation and the Riemann boundary problem on a torus. Phys. Lett. 92A, 51–55 (1982)
Nijhoff, F.W.: A higher-rank version of the Q3 equation (2011). arXiv:1104.1166
Nijhoff, F.W.: On some “Schwarzian equations’’ and their discrete analogues. In: Fokas, A.S., Gel’fand, I.M. (eds.) Algebraic Aspects of Integrable Systems: in Memory of Irene Dorfman, pp. 237–260. Birkhäuser, Basel (1996)
Nijhoff, F.W.: Discrete Painlevé equations and symmetry reduction on the lattice. In: Bobenko, A.I., Seiler, R. (eds.) Discrete Integrable Geometry and Physics, pp. 209–234. Oxford Univ. Press, Oxford (1999)
Nijhoff, F.W., Atkinson, J.: Elliptic N-soliton solutions of ABS lattice equations. Int. Math. Res. Not. 2010(20), 3837–3895 (2010)
Nijhoff, F.W., Capel, H.W., Wiersma, H.W., Quispel, G.R.W.: Bäcklund-transformations and 3-dimensional lattice equations. Phys. Lett. 105A, 267–272 (1984)
Nijhoff, F.W., Papageorgiou, V.G.: Lattice equations associated with the Landau–Lifshitz equations. Phys. Lett. 141A, 269–274 (1989)
Nijhoff, F.W., Papageorgiou, V.G., Capel, H.W., Quispel, G.R.W.: The lattice Gel’fand–Dikii hierarchy. Inverse Probl. 8, 597–621 (1992)
Nijhoff, F.W., Quispel, G.R.W., Capel, H.W.: Direct linearisation of nonlinear difference–difference equations. Phys. Lett. 97A, 125–128 (1983)
Nijhoff, F.W., Quispel, G.R.W., van der Linden, J., Capel, H.W.: On some linear integral equations generating solutions of nonlinear partial differential equations. Physica 119A, 101–142 (1983)
Rodin, Yu.L.: The Riemann boundary problem on a torus and the inverse scattering problem for the Landau–Lifschitz equation. Lett. Math. Phys. 7, 3–8 (1983)
Santini, P.M., Ablowitz, M.J., Fokas, A.S.: The direct linearization of a class of nonlinear evolution equations. J. Math. Phys. 25, 2614–2619 (1984)
Sun, Y.Y.: Applications of Elliptic Functions in Discrete Integrable Systems, PhD Thesis, Shanghai University (2017) (in Chinese)
Tongas, A.S., Nijhoff, F.W.: The Boussinesq integrable system. Compatible lattice and continuum structures. Glasgow Math. J. 47A, 205–219 (2005)
Unterkofler, K.: On the solutions of Halphen’s equation. Differ. Integr. Equ. 14, 1025–1050 (2001)
Walker, A.: Similarity Reductions and Integrable Lattice Equations. PhD Thesis, University of Leeds (2001). http://etheses.whiterose.ac.uk/7190/
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, 4th edn. Cambridge Univ. Press, Cambridge (1927)
Yoo-Kong, S., Nijhoff, F.W.: Elliptic \((N, N^{\prime })\)-soliton solutions of the lattice Kadomtsev–Petviashvili equation. J. Math. Phys. 54, 043511 (2013)
Zhang, D.J., Zhao, S.L., Nijhoff, F.W.: Direct linearisation of the extended lattice BSQ system. Stud. Appl. Math. 129, 220–248 (2012)
Acknowledgements
This project is supported by the NSF of China (Nos.11631007, 11875040, 12001369 and 12271334) and Shanghai Sailing Program (No. 20YF1433000). FWN acknowledges the hospitality of the Department of Mathematics of Shanghai University for various visits during which this paper was initiated and finalised. He is currently supported by EPRC grant EP/W007290/1.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H-T. Yau.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A: Formulae for elliptic functions
Here, we collect some useful formulae for elliptic functions, see also the standard textbooks e.g. [38]. The Weierstrass sigma-function is defined by
with \(\omega _{kl}=2k\omega + 2\ell \omega '\) and \(2\omega \), \(2\omega '\) being a fixed pair of the primitive periods. The relations between the Weierstrass functions are given by
where \(\sigma (x)\) and \(\zeta (x)\) are odd functions and \(\wp (x)\) is an even function of its argument. We recall also that the \(\sigma (x)\) is an entire function, and \(\zeta (x)\) is a meromorphic function having simple poles at \(\omega _{kl}\), both being quasi-periodic, i.e. obeying the periodicity conditions
in which the half periods satisfy the relation
whereas \(\wp (x)\) is doubly periodic. From an algebraic point of view, the most important property of these elliptic functions is the existence of a number of functional relations, the most fundamental being
which can also be cast into the following form
The well-known three-term relation for \(\sigma (x)\) is a consequence of (A.4)
and this equation can be cast into the following convenient form
which is obtained from the elliptic analogue of the partial fraction expansion, i.e. Eq. (A.5). Another important relation is given by
as well as
The well-known addition formula for the Weierstrass elliptic function can be written in the form
which together with the relation
characterises the addition rule on the elliptic curve parametrised by \((\wp (x),\wp '(x))\). Equation (A.10b) is a consequence of the determinant relation
which is a special case of the famous Frobenius-Stickelberger formula
which is an elliptic type van der Monde determinant.
An important special relation used in the derivations of Sects. 2.2 and 5.2 is:
which is used in the derivation of the various versions of the discrete KP equation for elliptic solutions. Furthermore, the following identity used several times in the BSQ reduction case:
which can be deduced by using (A.8)–(A.10a).
Appendix B: Derivation of the fundamental KP relations
Here we derive the fundamental relations of Sect. 2.2, Eqs. (2.10)–(2.15). Let \({{\varvec{C}}}\) be an infinite matrix as in (2.6), obeying the linear relations (2.8). Let us introduce the infinite matrix \({{\varvec{U}}}^0_\xi \) defined by the relation
where the infinite matrix \({\varvec{\varOmega }}_\xi \) is given by
where \(\varPhi _\xi (x)\) is the function (2.1). Eq. (B.2) should be viewed as a formal power series in both the operators \({\varvec{\varLambda }}\) and \(\,^{t\!}{\varvec{\varLambda }}\) where the normal ordering symbol \({:} ~{:}\) means that all operators \(\,^{t\!}{\varvec{\varLambda }}\) are ordered on the left of the projection matrix \({{\varvec{O}}}\) and all operators \({\varvec{\varLambda }}\) on the right of \({{\varvec{O}}}\). Then, as a consequence of the addition formula (A.7), we have the operator identity (2.7).
Since in (B.1) the dependence of \({{\varvec{U}}}^0_\xi \) only stems from the matrix \({\varvec{\varOmega }}_\xi \) and the dependence on the discrete variables n, m, l via the matrix \({{\varvec{C}}}\), we can derive the elementary shift relation of \({{\varvec{U}}}^0_\xi \) as follows:
and since from (B.1) we have
we arrive at:
Multiplying (B.3) from the left by \(\varPhi _\xi ({\varvec{\varLambda }})\) and from the right by \(\varPhi _{\xi +\delta }(\,^{t\!}{\varvec{\varLambda }})\) and setting
in accordance with (2.9), we get
The coefficients in the first and second term of (B.4) can be cast in the form of the functions \(-\chi ^{(1)}_{-\delta ,\xi +\delta }(\,^{t\!}{\varvec{\varLambda }})\) and \(\chi ^{(1)}_{\delta ,\xi }({\varvec{\varLambda }})\) respectively.
Next, we derive the relations for infinite vectors
with \(\rho _\kappa \) and \(\sigma _{\kappa '}\) obeying (2.3). Deriving the shift relation for \({\varvec{u}}^0_\kappa (\xi )\) proceeds as follows
where in the third step we have made use of (B.3). Setting
we obtain from the latter result
where the coefficient in the first term on the r.h.s. equals \(\chi _{\delta ,\xi }^{(1)}({\varvec{\varLambda }})\). In a similar way the shift relation for the adjoint vector \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }^0(\xi )\) can be derived, namely
Multiplying the latter result from the right by \(\varPhi _{\xi +\delta }(\,^{t\!}{\varvec{\varLambda }})\) we get
where the coefficient in the first terms on the r.h.s. is identified with \(-\chi ^{(1)}_{-\delta ,\xi }(\,^{t\!}{\varvec{\varLambda }})\).
Finally, we derive the relations from the \(\tau \)-function defined by (2.10). Applying the \(\widetilde{~}\) -shift we have
where we have subsequently used the shift relation (2.8a), the relation (2.7) for the Cauchy kernel and the invariance of the determinant under similarity transformations, as well as the definition (B.1). Thus, we get
where the latter step uses the fact that \({{\varvec{O}}}\) is a rank 1 projector matrix and we apply the famous Weinstein-Aroszajn formula:
for determinants involving a rank 1 perturbation from the unit matrix.
Appendix C: Some higher-order elliptic identities
The following higher addition rules were established in Appendix C of [7], which for the sake of self-containedness we reiterate here. They may be used, in future, for deriving the higher reductions from the lattice KP system to the lattice GD hierarchy, and they play a role in establishing the defining relations for the higher elliptic roots of unity. First, we have the following generalisation of (A.4)
which derives from:
Furthermore, we have [7]
At the next level we get the following identity:
Setting \(w=z+\delta \) and taking the limit \(\delta \rightarrow 0\) in (C.2) we obtain:
which, after a renaming of the arguments, can be cast into the following form:
In the next step we set \(z=y+\varepsilon \) in (C.3), and take the limit \(\varepsilon \rightarrow 0\), which yields the relation
which generalises (C.1). In the final step, however, setting \(y=x+\gamma \) and letting \(\gamma \rightarrow 0\) (which requires expansions up to third order in \(\gamma \)), we get the identity
The previous relations can be obviously generalised to arbitrary order. Thus, the general form of the basic identity (A.7) (3-term relation for the \(\sigma \)-function, or the elliptic partial fraction expansion formula) is:
Extending this identity to \(n+1\) variables, including a \(\kappa _0\) and \(x_0\), and subsequently taking the limit \(x_0=x_1+\varepsilon \), with \(\varepsilon \rightarrow 0\), we obtain the following identity (after some obvious relabelling of parameters and changes of variables):
These identities are associated with the Frobenius-Stickelberger determinantal formula (A.12).
1.1 Appendix D: Lax pairs
The Lax pairs for the BSQ type lattice equations are derived from the fundamental system of relations (4.17b) and (4.17d), or equivalently (4.19b) and (4.19d), for the vectors \({{\varvec{u}}}_\kappa (\xi )\) together with (4.18b), and their counterparts in the other lattice direction. By choosing specific components of these infinite vectors we construct 3-component vectors which will yield the eigenvectors for the corresponding 3\(\times \)3 matrix Lax pairs. There are two different choices of entries that we will consider, leading to the Lax pairs for the lattice BSQ system and lattice modified BSQ system respectively.
a) Lax pair for the lattice BSQ system
In this case we specify the entries:
and consider the relations (4.17), together with (4.18), to derive a Lax pair for the 3-component vector
In fact, the dynamical relations (4.17), using the above notation, can be rewritten as:
and similar relations for the other lattice direction. Taking the 0-component of these relations and their counterparts in the other direction we constitute the Lax pair as the shift relations for the vector \({\varvec{\phi }}(\xi )\), as given by
with Lax matrices \({\varvec{L}}_\kappa (\xi )\) and \({\varvec{M}}_\kappa (\xi )\) given by
in which
and \({\varvec{M}}_\kappa (\xi )\) similarly replacing \(\widetilde{~}\) by \(\widehat{~}\) and \(\delta \) by \(\varepsilon \) everywhere. The entry \(\clubsuit \) is such that \(\det ({\varvec{L}}_\kappa (\xi ))=-\tfrac{1}{2}(\wp '(\delta )+\wp '(\kappa ))\) . The compatibility of the Lax pair, \( \widetilde{{\varvec{L}}_\kappa (\xi )}{\varvec{M}}_\kappa (\xi )= \widehat{{\varvec{M}}_\kappa (\xi )} {\varvec{L}}(\kappa (\xi )\) , yields the following coupled set of relations
where a number of identities are used, such as (5.4a) and (5.4b), as well as
This coupled set of equations is similar to the one appearing in [25, 35] and by elimination of the quantities \(u_{1,0}\) and \(u_{0,1}\) yields the lattice BSQ system (5.6). In principle, the Lax pair can be used to investigate initial-boundary value problems for the lattice BSQ equation in a similar way as the inverse scattering scheme for the BSQ equation. That approach to solutions is beyond the scope of the present paper.
b) Lax pair for the lattice modified BSQ system
In the case of the lattice modified BSQ equation we can chose an eigenvector in the form
depending on the additional parameter \(\alpha \). For this choice of eigenvector we can make use of the identities (5.8) and (5.9a) to derive the corresponding relations for the entries of \(\psi _\kappa (\xi )\) from the fundamental relations (4.17b) and (4.17d). Thus, we can derive the Lax pairs
where the modified BSQ Lax matrices are given by
in which
and similarly \({\mathcal {M}}_\kappa \) obtained from (D.6b) by replacing \(\delta \) by \(\varepsilon \) (and hence \(p_\xi \) by \(q_\xi \)) and \(\widetilde{~}\) by \(\widehat{~}\). The quantity indicated by \(\spadesuit \) is such that \(\det ({\mathcal {L}}_\kappa )=-\tfrac{1}{2}(\wp '(\delta )+\wp '(\kappa ))\widetilde{v_\alpha }/v_\alpha \) and \(\det ({\mathcal {M}}_\kappa )=-\tfrac{1}{2}(\wp '(\varepsilon )+\wp '(\kappa ))\widehat{v_\alpha }/v_\alpha \) .
The compatibility relation \(\widehat{{\mathcal {L}}_\kappa }\,{\mathcal {M}}_\kappa =\widetilde{{\mathcal {M}}_\kappa }\,{\mathcal {L}}_\kappa \) leads to the relations (5.19a), (5.17a) and (5.20a), as well as to the relation
which can be obtained from combining (5.16a) and (5.15a). The relations thus obtained from the Lax pair allow us to reconstruct the lattice modified BSQ equation (5.21). Similarly, a Lax pair can be set up for the modified BSQ equation in the form (5.22), starting from the relations in Sect. 4 for the adjoint vector \(\,^{t\!}{{\varvec{u}}}_{\kappa ^\prime }(\xi )\), and involving the quantities \(w_\beta \). We will omit the details here.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Nijhoff, F.W., Sun, Yy. & Zhang, Dj. Elliptic Solutions of Boussinesq Type Lattice Equations and the Elliptic \(N{\textrm{th}}\) Root of Unity. Commun. Math. Phys. 399, 599–650 (2023). https://doi.org/10.1007/s00220-022-04567-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-022-04567-8