Abstract
We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to its constituting lines. There is a subclass of “pure line soliton solutions” for which we find that, in this limit, the distribution of polarizations is fully determined by a Yang–Baxter map. For a vector KP equation, this map is given by an R-matrix, whereas it is a nonlinear map in the case of a more general matrix KP equation. We also consider the corresponding Korteweg–deVries reduction. Furthermore, exploiting the fine structure of soliton interactions in the tropical limit, we obtain an apparently new solution of the tetrahedron (or Zamolodchikov) equation. Moreover, a solution of the functional tetrahedron equation arises from the parameter dependence of the vector KP R-matrix.
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1 Introduction
A line soliton solution of the scalar Kadomtsev–Petviashvili (KP-II) equation (see, e.g., [16]) is, at fixed time t, an exponentially localized wave on a plane. The “tropical limit” takes it to a piecewise linear structure, a planar graph that represents the wave crest, with values of the dependent variable attached to its edges. Via the Maslov dequantization formula (also used in “ultra-discretization” [22]), the tropical limit graph at fixed t can be conveniently computed as the boundary of “dominating phase regions” in the (xy) plane (see Sect. 4). Applications of this method in the context of integrable PDEs can be found in [3, 4, 6,7,8, 14, 19], for example.
In this work, we consider the \(m \times n\)matrix potential KP equation
where K is a constant \(n \times m\) matrix and \(\phi \) an \(m \times n\) matrix, depending on independent variables x, y, t, and a subscript indicates a corresponding partial derivative. We will refer to this equation as \(\hbox {pKP}_K\).
If \(\phi \) is a solution of (1.1), then \(\phi _R := \phi K\) and \(\phi _L := K \phi \) solve the ordinary \(m \times m\), respectively, \(n \times n\), matrix potential KP equation. We also note that, if \(K = T K' S\) with a constant \(m' \times m\) matrix S and a constant \(n \times n'\) matrix T, then the \(m' \times n'\) matrix \(\phi ' = S \phi T\) satisfies the \(\hbox {pKP}_{K'}\) equation, as a consequence of (1.1).
In the vector case \(n=1\), writing \(K = (k_1,\ldots ,k_m)\) and \(\phi = (\phi _1, \ldots ,\phi _m)^\intercal \), (1.1) becomes the following system of coupled equations,
By choosing \(T=1\) and any invertible \(m \times m\) matrix S that has K as its first row, we have \(K = K' S\) with \(K' = (1,0,\ldots ,0)\). In terms of the new variable \(\phi ' = S \phi \), the above system thus consists of one scalar pKP equation and \(m-1\) linear equations involving the dependent variable of the former.
For
we obtain from (1.1) the \(m \times n\) matrix KP equation
The extension of the scalar KP equation to a matrix version achieves that solitons carry internal degrees of freedom. The value of the dependent variable along a segment of the suitably defined (piecewise linear) tropical limit graph will be referred to as “polarization” in the following.
The Korteweg–deVries (KdV) reduction of (1.2) is
which we will refer to as \(\hbox {KdV}_K\). If K is the identity matrix, this is the matrix KdV equation (see, e.g., [11]). The 2-soliton solution of the latter yields a map from polarizations at \(t \ll 0\) to polarizations at \(t \gg 0\). It is known [12, 24] that this yields a Yang–Baxter map, i.e., a set-theoretical solution of the (quantum) Yang–Baxter equation (also see [1, 23] for the case of the vector Nonlinear Schrödinger equation). Not surprisingly, this is a feature preserved in the tropical limit. The surprising new insight, however, is that this map governs the evolution of polarizations throughout the tropical limit graph of a soliton solution. In case of a vector KdV equation, i.e., \(\hbox {KdV}_K\) with \(n=1\), it is given by an R-matrix, a linear map solution of the Yang–Baxter equation.
More generally, we will explore in this work the tropical limit of “pure” (see Sect. 3) soliton solutions of the above K-modified matrix KP equation and demonstrate that a Yang–Baxter map governs their structure. There are lots of soliton solutions beyond pure solitons (see, e.g., [16] for the scalar case), but for them a Yang–Baxter map is no longer sufficient to describe the behavior.
In the case of the vector KP equation, the expression for a pure soliton solution involves a function \(\tau \) which is a \(\tau \)-function of the scalar KP equation. Its tropical limit at fixed t determines a planar graph, and the vector KP soliton solution associates in this limit a constant vector (polarization) with each linear segment of the graph. The polarization values are then related by a linear Yang–Baxter map, represented by an R-matrix, which does not depend on the independent variables x, y, t, but only on the “spectral parameters” of the soliton solution.
Section 2 summarizes a binary Darboux transformation for the \(\hbox {pKP}_K\) equation and applies it to a trivial seed solution in order to obtain soliton solutions. In Sect. 3, we restrict out consideration to the subclass of “pure” soliton solutions. This essentially disregards solutions with substructures of the form of Miles resonances. Section 4 addresses the tropical limit of pure soliton solutions. The cases of two and three solitons are then treated in Sects. 5 and 6. Section 7 provides a general proof of the fact that, in the vector case, an R-matrix relates the polarizations at crossings. The linearity of the Yang–Baxter map in the vector case is certainly related to the particularly simple structure of the vector pKP equation mentioned above. In Sect. 8 we show how to construct a pure N-soliton solution of the vector KP equation from a pure N-soliton solution of the scalar KP equation, N vector data and the aforementioned R-matrix. Section 9 extends our exploration of the vector KP 3-soliton case and presents an apparently new solution of the tetrahedron (Zamolodchikov) equation (see, e.g., [9] and references cited there). In Sect. 10, we reveal the structure of the vector KP R-matrix, which leads us to a more general two-parameter R-matrix. Its parameter dependence determines, via a “local” Yang–Baxter equation [18] (also see [9]), a solution of the functional tetrahedron equation (see, e.g., [9, 15, 20]), i.e., the set-theoretical version of the tetrahedron equation. Finally, Sect. 11 contains some concluding remarks.
2 Soliton solutions of the K-modified matrix KP equation
The following describes a binary Darboux transformation for the \(\hbox {pKP}_K\) Eq. (1.1). This is a simple extension of what is presented in [5], for example. Let \(\phi _0\) be a solution of (1.1). Let \(\theta \) and \(\chi \) be \(m \times N\), respectively, \(N \times n\), matrix solutions of the linear equations
Then the system
is compatible and can thus be integrated to yield an \(N \times N\) matrix solution \(\Omega \). If \(\Omega \) is invertible, then
is a new solution of (1.1).
For vanishingFootnote 1 seed solution, i.e., \(\phi _0=0\), soliton solutions are obtained as follows. Let
where \(P_a, Q_b\) are constant \(N \times N\) matrices, \(\theta _a, \chi _b\) are constant \(m \times N\), respectively, \(N \times n\) matrices, and
If, for all a, b, the matrices \(P_a\) and \(Q_b\) have no eigenvalue in common, there are unique \(N \times N\) matrix solutions \(W_{ba}\) of the Sylvester equations
Then (2.1) is solved by
with a constant \(N \times N\) matrix \(\Omega _0\), and (2.2) determines a soliton solution of (1.1) [and thus via \(u = 2 \, \phi _x\) a solution of (1.2)], if \(\Omega \) is everywhere invertible.
Remark 2.1
Corresponding solutions of the \(\hbox {pKP}_K\)hierarchy are obtained by replacing (2.3) with \(\vartheta (P) = \sum _{r=1}^\infty t_r \, P^r\), where \(t_1=x\), \(t_2=y\), \(t_3=t\). \(\square \)
3 Pure soliton solutions
In the following, we restrict our considerations to the case where \(A=B=1\). Then there remains only a single Sylvester equation,
Moreover, we will restrict the matrices \(P_1\) and \(Q_1\) to be diagonal. It is convenient to name the diagonal entries (“spectral parameters”) in two different ways,
We further write
where \(\xi _i\) are m-component column vectors and \(\eta _i\) are n-component row vectors. Then the solution of the above Sylvester equation is given by
Furthermore, we set \(\Omega _0 = I_N\), the \(N \times N\) identity matrix. Hence \(\Omega = (\Omega _{ij})\) with
where \(\delta _{ij}\) is the Kronecker delta. We call soliton solutions obtained from (2.2), with the above restrictions, “pure solitons”. All what follows refer to them.
We introduce
Instead of using \((a_1,\ldots ,a_N)\) as a subscript (or superscript), we will simply write \(a_1 \ldots a_N\) in the following. For example, \(\vartheta _{a_1 \ldots a_N} = \vartheta _{(a_1,\ldots ,a_N)}\).
From (2.2), we find that the pure soliton solutions of the \(\hbox {pKP}_K\) equation are given by
with
where \(\mathrm {adj}(\Omega )\) denotes the adjugate of the matrix \(\Omega \) and \(\varvec{2} := 2\ldots 2 = (2,\ldots ,2)\).
Proposition 3.1
\(\tau \) and F have expansions
with constants \(\mu _I\) and constant \(m \times n\) matrices \(M_I\), where \(\mu _{\varvec{2}}=1\) and \(M_{\varvec{2}}=0\).
Proof
From the definition of the determinant, \(\det \Omega = \epsilon ^{i_1 \ldots i_N} \, \Omega _{1 i_1} \cdots \Omega _{N i_N}\), with the Levi–Civita symbol \(\epsilon ^{i_1 \ldots i_N}\) and summation convention, we know that \(\det \Omega \) consists of a sum of monomials of order N in the entries \(\Omega _{ij}\). Here the latter is given by (3.1). If no diagonal term \(\Omega _{ii}\) appears in a monomial, its phase factor is \(\hbox {e}^{\vartheta _{\varvec{1}} - \vartheta _{\varvec{2}}}\). If one diagonal entry \(\Omega _{ii} = 1 + w_{ii} \, \hbox {e}^{\vartheta (p_{i,1}) - \vartheta (p_{i,2})}\) appears in a monomial, the latter splits into two parts. Only the part arising from the summand 1 is different as now the phase factor is \(\hbox {e}^{\vartheta _{\varvec{1}} - \vartheta _{\varvec{2}} - \vartheta (p_{i,1}) + \vartheta (p_{i,2})}\). From monomials containing several diagonal entries of \(\Omega \), we obtain summands with a phase factor of the form
Finally, from a monomial with N diagonal entries of \(\Omega \), we also obtain a constant term, namely 1. Now our assertion (3.5) follows since \(\tau \) is \(\det \Omega \) multiplied by \(\hbox {e}^{\vartheta _{\varvec{2}}}\). Clearly, \(\mu _{\varvec{2}}=1\).
According to the Laplace (cofactor) expansion \(\det \Omega = \sum _{j=1}^N \Omega _{ij} \, \mathrm {adj}(\Omega )_{ji}\) with respect to the ith row, the term \(\Omega _{ij} \, \mathrm {adj}(\Omega )_{ji}\) consists of all summands in \(\det \Omega \) having \(\Omega _{ij}\) as a factor. (3.5) implies that a summand of \(\hbox {e}^{\vartheta _{\varvec{2}}} \mathrm {adj}(\Omega )_{ji}\) then has a phase factor of the form \(\hbox {e}^{\vartheta _I - \vartheta (p_{j,1}) + \vartheta (p_{i,2})}\), with some \(I \in \{1,2\}^N\), so that \(\hbox {e}^{\vartheta _{\varvec{2}}} \, (\hbox {e}^{\vartheta (P_1)} \, \mathrm {adj}(\Omega ) \, \hbox {e}^{-\vartheta (Q_1)})_{ji}\) has the phase factor \(\hbox {e}^{\vartheta _I}\). Hence (3.6) holds. Furthermore, no entry of \(\hbox {e}^{\vartheta (P_1)} \, \mathrm {adj}(\Omega ) \, \hbox {e}^{-\vartheta (Q_1)}\) is constant, and hence, \(M_{\varvec{2}}=0\). \(\square \)
Remark 3.2
The introduction of the redundant factor \(\hbox {e}^{\vartheta _{\varvec{2}}}\) in (3.2), via the definitions (3.3) and (3.4), achieves that \(\tau \) and F are linear combinations of exponentials \(\hbox {e}^{\vartheta _I}\), \(I \in \{1,2\}^N\), in which case we have a very convenient labeling. This is also so if we choose the factor \(\hbox {e}^{-\vartheta _{\varvec{1}}}\) instead, which leads to an expansion in terms of \(\hbox {e}^{-\vartheta _I}\), now with \(M_{\varvec{1}}=0\). \(\square \)
Regularity of a pure soliton solution requires \(\mu _I \ge 0\) for all \(I \in \{1,2\}^N\) (or equivalently \(\mu _I \le 0\) for all \(I \in \{1,2\}^N\)) and \(\mu _I \ne 0\) for at least one I. If \(\mu _I =0\) for some I, this means that the phase \(\vartheta _I\) is not present in the expression for \(\tau \). In this case, one has to arrange the data in such a way that \(M_I =0\) in order to avoid unbounded exponential growth of the soliton solution in some phase region. But we will disregard such cases and add the condition \(\mu _I > 0\), \(\forall I \in \{1,2\}^N\), to our definition of pure soliton solutions.
It follows that the corresponding solution of the KP equation is given by
where
Using Jacobi’s formula for the derivative of a determinant, we obtain
which implies
and thus
Using (3.2) in (3.7), and reading off the coefficient of \(\hbox {e}^{\vartheta _I}\), we find
Remark 3.3
If \(n=1\), (3.7) reads
and
is a solution of the scalar KP equation. If \(n>1\), \(\mathrm {tr}(K u)\) is not in general a solution of the scalar KP equation. \(\square \)
Remark 3.4
Dropping the redundant factor \(\hbox {e}^{\vartheta _{\varvec{2}}}\) in (3.3) and (3.4) means that we have to multiply the above expressions (3.5) and (3.6) for \(\tau \) and F by \(\hbox {e}^{-\vartheta _{\varvec{2}}}\). It is then evident that \(\phi \) only depends on differences of phases of the form \(\vartheta (p_{i,1}) - \vartheta (p_{i,2}) = (p_{i,1} - p_{i,2}) \, x + (p_{i,1}^2 - p_{i,2}^2) \, y + (p_{i,1}^3 - p_{i,2}^3) \, t\). As a consequence, setting \(p_{i,2} = - p_{i,1}\), i.e., \(q_i = -p_i\), eliminates the y terms in all phases. This means that under this condition for the parameters, we could have started as well with \(\vartheta (P) = x \, P + t \, P^3\), hence without the y term in (2.3). In this way, contact is made with the \(\hbox {KdV}_K\) reduction in \(\hbox {KP}_K\). \(\square \)
4 Tropical limit of pure soliton solutions
A crucial point is that we define the tropical limit of the matrix soliton solution via the tropical limit of the scalar function \(\tau \) (cf. [6,7,8]). Let
In a region where a phase \(\vartheta _I\) dominates all others, in the sense that \(\log (\mu _I \, \hbox {e}^{\vartheta _I}) > \log (\mu _J \, \hbox {e}^{\vartheta _J})\) for all participating \(J \ne I\), the tropical limit of the potential \(\phi \) is given by (4.1). It should be noticed that these expressions do not depend on the coordinates x, y, t.
The boundary between the regions associated with the phases \(\vartheta _I\) and \(\vartheta _J\) is determined by the condition
Not all parts of such a boundary are visible at fixed time, since some of them may lie in a region where a third phase dominates the two phases. The tropical limit of a soliton solution at a fixed time t has support on the visible parts of the boundaries between the regions associated with phases appearing in \(\tau \). On such a visible boundary segment, the value of u is given by
For \(I = (a_1,\ldots ,a_N)\) we set
At fixed time, the set of line segments associated with the kth soliton are obtained from (4.2) with \(I = I_k(1)\) and \(J=I_k(2)\), for all possible I. They satisfy
All these line segments have the same slope \(-(p_{k,2}+p_{k,1})^{-1}\) in the xy plane, and hence, they are parallel. The shifts between them are given by
They give rise to the familiar asymptotic “phase shifts” of line solitons. The tropical limit of u on a visible line segment of the kth soliton is given by
The value of u at a visible triple phase coincidence is
Instead of the above expressions for the tropical values of u we will rather consider
which has the form of a discrete derivative. Since (3.8) and (4.1) imply
the latter values are normalized in the sense that
If \(I=(a_1,\ldots ,a_N)\) and \(i \ne j\), let
The normalized tropical values of u satisfy
These identities are simply consequences of the definition of \(\hat{u}_{IJ}\). They linearly relate the (normalized) polarizations at points of the tropical limit graph, where three lines meet.
5 Pure 2-soliton solutions
Let \(N=2\). Then we have
where
and
The tropical values of the \(\hbox {pKP}_K\) solution \(\phi \) in the dominant phase regions are then given by
Remark 5.1
The above values \(\phi _{ab}\) solve the following nonlinear equation,
Addressing more than two solitons, nonzero counterparts of \(\phi _{22}\) will show up, as displayed in this equation. \(\square \)
For the tropical values of \(\hat{u}\) along the phase region boundaries, we obtain
where \(1_m\) stands for the \(m \times m\) identity matrix. For the in/out classification, see Fig. 1 below. All the matrices in (5.2) have rank one, which is not at all obvious from the form of \(\phi _{ab}\). We obtain the following nonlinear relation between “incoming” and “outgoing” polarizations,
where
We note that \(\alpha \, \alpha _{\mathrm {in}} = 1\). (5.3) determines a new nonlinear Yang–Baxter map
with parameters \(p_i,q_i\), \(i=1,2\). We verified directly that this satisfies the Yang–Baxter equation
where the subscripts indicate on which two factors of a threefold Cartesian product the map \(\mathcal {R}\) acts. An explanation why the Yang–Baxter equation holds will be provided in Sect. 6.
Writing
determines \(\xi _{1,\mathrm {in/out}}\) and \(\eta _{1,\mathrm {in/out}}\) up to scalings. We find
and
is another form of the above Yang–Baxter map.
Remark 5.2
In (5.2), we found that \(u_{2,\mathrm{in}}\) and \(u_{1,\mathrm{out}}\) have a simple elementary form. They are the polarizations at the two boundary lines of the dominating phase region numbered by \(22 = (2,2)\), see Fig. 1. We know from Proposition 3.1 that it is special since \(M_{22}=0\). Considering an “evolution” in negative x direction (instead of y direction), thus, offers a more direct derivation of the Yang–Baxter map. \(\square \)
Example 5.3
Let \(m=3\) and \(n=2\). Choosing
and
we obtain the first contour plot, at \(t=0\), shown in Fig. 1. Figure 2 shows plots of the components of the transpose of u. Choosing instead \(p_2\) close to \(q_1\) reveals an “inner structure” of crossings, see the second plot in Fig. 1. This is the (phase) shift mentioned in Sect. 4. \(\square \)
Remark 5.4
We should stress that the relevant structures are actually three-dimensional and our figures only display a two-dimensional cross section. Instead of displaying structures in the xy plane at constant t, we may as well look at those in the xt plane at constant y. The latter becomes relevant if we consider the \(\hbox {KdV}_K\) reduction. \(\square \)
Remark 5.5
Soliton solutions of \(\hbox {KdV}_K\) are obtained from those of \(\hbox {KP}_K\) by setting \(q_i = - p_i\), \(i=1,\ldots ,N\), see Remark 3.4. Then the above equations reduce to
with
If K is the \(N \times N\) identity matrix, this becomes the Yang–Baxter map first found by Veselov [24], also see [12, 21]. The factor \(\alpha _{\mathrm {in}}^{-1/2}\) is missing in these publications, but such a factor is necessary for the map to satisfy the Yang–Baxter equation. It should be noticed that Veselov’s “Lax pair” [24] only determines the Yang–Baxter map up to a factor, but such a factor has to be chosen appropriately in order to satisfy the Yang–Baxter equation. One can avoid the square root in the above expression at the price of having an asymmetric appearance of factors \(\alpha _{\mathrm {in}}^{-1}\). \(\square \)
5.1 Pure column vector 2-soliton solutions
We set \(n=1\). Now the \(\eta _i\) are scalars and drop out of the relevant formulas. Introducing
we have
and thus
Generalizing the matrix that appears on the right-hand side to
and letting this act from the right on the ith and jth slot of a threefold direct sum, the Yang–Baxter equation holds. This can be checked directly or inferred from a 3-soliton solution, see Sect. 6.
Remark 5.6
The reduction to vector \(\hbox {KdV}_K\) via \(q_i = - p_i\) (see Remark 3.4) leads to
This rules the evolution of initial polarizations (at \(t \ll 0\)) step by step along the tropical limit graph in two-dimensional space-time. The R-matrix (5.7) also describes the elastic collision of non-relativistic particles with masses \(p_i\) in one dimension, see [17]. \(\square \)
5.2 Pure row vector 2-soliton solutions
Now we set \(m=1\). Then the \(\xi _i\) are scalars and drop out of the relevant formulas. Introducing
we have
so that
which determines a Yang–Baxter map. Let
act on the ith and jth slot of a direct sum. Then the Yang–Baxter equation holds. We note that \(\tilde{R}(p_i,q_i;p_j,q_j) = R(q_i,p_i;q_j,p_j)^\intercal \).
6 Pure 3-soliton solutions
For \(N=3\) we find
where again \(\kappa _{ij} = \eta _i K \xi _j\), and
Furthermore, we obtain
where
Note that \(\alpha _{ij} = \alpha _{ijj}\). Recall that the coefficient of \(\hbox {e}^{\vartheta _{abc}}\) in the expression for \(\tau \), respectively, F, has been named \(\mu _{abc}\), respectively, \(M_{abc}\). The tropical value in the region where \(\vartheta _{abc}\) dominates all other phases is given by
The corresponding values can be read off from (6.1) and (6.2).
The Yang–Baxter property of the nonlinear map (5.4) can be deduced from the pure 3-soliton solution in the following way. Numbering the (in y direction) incoming solitons by 1, 2, 3 in x direction, for \(t \ll 0\) first (according to increasing values of y) solitons 1 and 2 interact, then solitons 1 and 3, and finally solitons 2 and 3. For \(t \gg 0\) solitons 2 and 3 meet first, then solitons 1 and 3, and finally solitons 1 and 2. Also see Fig. 3 below. Recalling that the polarizations along the tropical limit graph do not depend on the variables x, y, t, this implies that in both cases we obtain the same outgoing polarizations. Hence, the Yang–Baxter equation (5.5) holds. This is worked out in detail only for the simpler vector KP case in the next subsection. But we checked the general case as well.
That we can check the Yang–Baxter equation in this way is because of the fact that, in the tropical limit and at fixed t, we have well-defined interaction points of solitons. In the original wave description, we can only compute the asymptotics, i.e., the structure of incoming and outgoing solitons, but we have no description of what happens in the interaction region.
Because of the exponential localization of waves, a pure N-soliton solution, \(N>2\), looks like a 2-soliton solution close enough to a crossing in the tropical limit graph. This becomes exact in the tropical limit. It implies that the Yang–Baxter map (5.4) acts at any crossing of the tropical limit graph. We will discuss this in more detail for the vector KP case in Sect. 7.
6.1 Pure vector KP 3-soliton solutions
Now we restrict our considerations to the vector case \(n=1\). Using
we obtain
The contour plots in Fig. 3 show the structure at fixed t with \(t < 0\) and \(t > 0\), respectively. The lines extending to the bottom are numbered by 1, 2, 3 from left to right (displayed as blue, red, green, respectively). Thinking of three particles undergoing a scattering process in y direction, they carry polarizations that change at crossings. As y increases we have
for \(t<0\), where the ith entry contains the polarization of the ith particle, and
for \(t>0\). The numbers ij assigned to the steps refer to the “particles” involved. In both cases, we start and end with the same vectors, and this implies the Yang–Baxter equation for the associated transformations. Let \(V_{a_1a_2a_3,b_1b_2b_3}\) be the column vector formed by the coefficients of \(\hat{u}_{a_1a_2a_3,b_1b_2b_3}\) with respect to \(\hat{\xi }_1,\hat{\xi }_2,\hat{\xi }_3\). The following matrices are composed of these column vectors,
They represent the triplets of polarizations constituting the above chains. Next we define matrices
which turn out to be given in terms of the R-matrix (5.6). For example,
The Yang–Baxter equation reads
Figure 4 shows plots of Ku for a choice of the parameters.
6.2 Vector KdV 3-soliton solutions
We impose the KdV reduction, see Remark 3.4, and replace (2.3) by
The additional last term introduces the next evolution variable s of the KdV hierarchy, also see Remark 2.1.
Let us consider, for simplicity, the \(\hbox {KdV}_K\) equation with \(m=3\) and \(K = (1,1,1)\), and the special solution with parameters
The tropical limit graph is displayed in Fig. 5 for \(p_1=1/2\), \(p_2=3/4\), \(p_3=1\), and different values of s. We have the matrix
of initial polarizations. The next values \(\hat{u}_{abc,def}\) are then obtained by application of the R-matrix (5.7) from the right, and so forth, following either the left or the right graph in Fig. 5 in upwards (i.e., t) direction. Since the initial and the final polarizations are the same, the Yang–Baxter equation holds. Here the R-matrix describes the time evolution of polarizations in the tropical limit.
Figure 5 suggests to think of R-matrices as being associated with bounded lines, which may be thought of as representing “virtual solitons”. Interaction of two solitons then means exchange of a virtual soliton, frequently called a “resonance”.
At \(s=0\), see the plot in the middle of Fig. 5, something peculiar occurs, namely a sort of three-particle interaction. This is a degenerate special case to which the Yang–Baxter description does not apply. To be precise, the statement that the Yang–Baxter map rules the polarizations along the tropical limit graph thus holds for \(s\ne 0\).
To understand what this means, let us more generally think of any system, depending continuously on a parameter, say s, and carrying a structure, which is described by a map such that the left-hand side of the Yang–Baxter equation is realized for \(s<0\), and the right-hand side for \(s>0\). Then there is a “transition point”, \(s=0\), where the Yang–Baxter equation does not apply. The system is actually more complete, since it also contains a transition structure. The latter, however, cannot be resolved into a sequence of three applications of a map. This is the situation we meet in the tropical limit analysis of pure KP multi-soliton solutions. There are isolated parameter values, corresponding to “transitions”, at which the Yang–Baxter equation does not apply.
7 Tropical limit of pure vector solitons and the R-matrix
We set \(n=1\) (vector case). The following results describe what happens at a crossing of two solitons, numbered by i and j, depicted as a contour plot in Fig. 6.
Lemma 7.1
Proof
This is quickly verified for the 2-soliton solution (\(N=2\)). But at a crossing, a general solution \(\phi \) is equivalent to a 2-soliton solution, since there the four elementary phases \(\vartheta (p_{i,a}),\vartheta (p_{j,a})\), \(a=1,2\), dominate all others, and hence, the exponential of any other phase vanishes in the tropical limit. \(\square \)
Remark 7.2
(7.1) can be regarded as a vector version of a scalar linear quadrilateral equation, satisfying “consistency on a cube”, see (15) in [2]. Such a linear relation does not hold in the matrix case where \(m,n>1\). But (5.1), with \(\phi _{ab}\) replaced by \(\phi _{I_{ij}(a,b)}\) (in which case we have \(\phi _{I_{ij}(2,2)} \ne 0\), in general) is a nonlinear counterpart of (7.1). The latter can be deduced from it for \(n=1\) by using (4.5). \(\square \)
Theorem 7.3
with
Proof
Using (4.4) we can directly verify that the following relations hold as a consequence of (7.1),
In matrix form, this is (7.2). \(\square \)
We already know that \(R(p_i,q_i;p_j,q_j)\) satisfies the Yang–Baxter equation.
8 Construction of pure vector KP soliton solutions from a scalar KP solution and the R-matrix
Given a \(\tau \)-function for a pure N-soliton solution of the scalar KP equation, the Yang–Baxter R-matrix found above can be used to construct a pure N-soliton solution of the vector KP equation. We will explain this for the case \(N=3\).
The \(\tau \)-function of the pure 3-soliton solution of the scalar KP-II equation is given by
as obtained by the Wronskian method (see, e.g., [13]). Comparison with (3.5) shows that \(\mu _I = \Delta _I\).
Starting at the bottom of both graphs in Fig. 7, we associate a column vector \(\xi _i\) with the ith soliton (counted from left to right) and normalize it such that \(K \xi _i=1\). Accordingly, we set
By consecutive application of the R-matrix (5.6), we find the polarizations on the further line segments, proceeding in y direction. There are different ways to proceed, but they are consistent since R satisfies the Yang–Baxter equation. For example,
This leads to
By “integration” of (4.4), we find \(\phi _{abc}\) up to a single constant. Setting \(\phi _{222}=0\), we obtain
From (4.1) we can read off \(M_{abc}\) and thus obtain via (3.2) and (3.4) the solution
of the vector pKP equation. This procedure can easily be applied to a larger number of solitons.
Remark 8.1
The construction described in this section cannot be extended to the matrix KP case, since then the function \(\tau \) does not correspond to a solution of the scalar KP equation, or of any other meaningful equation.
9 A solution of the tetrahedron equation
In this section, we address again the case of three pure solitons, see Sect. 6. Because of the occurrence of phase shifts, in the tropical limit a crossing of two solitons does not really take place in a point. Figure 8 shows this for the second interaction (in the vertical y direction) in Fig. 3, or Fig. 7.
We may think of associating with the additional edge in the left plot (\(t<0\)) the polarization of the boundary between the phase regions numbered by \(121 = (1,2,1)\) and 222, and in the right plot (\(t>0\)) that of the boundary between the phases 112 and 211. But this does not lead us to something meaningful. Instead, we make an educated guess and associate a mean value of vectors with an additional edge,
and
The inner boundary lines shown in the two plots in Fig. 8 are now associated with \(V_2\), respectively, \(\hat{V}_2\). In the first plot, the boundary between phase regions 122 and 221 is hidden, but there is also a polarization associated with it, namely \(V_{122,221}\). \(V_2\) is the mean value of the latter and \(V_{121,222}\), which belongs to the visible inner boundary segment.
Now we have
where \(\gamma ^{ijk} = (\gamma ^{ijk}_{rs})\) with
and
Here we set
Let \(S_{\alpha \beta \gamma }(i,j,k)\) be the \(6 \times 6\) matrix which acts via S(i, j, k) on the positions \(\alpha ,\beta ,\gamma \) of a 6-component column vector, and as the identity on the others. Let
We verified that this constitutes a (to our knowledge new) solution of the tetrahedron (Zamolodchikov) equation (see, e.g., [9] and references cited there)
An explanation for the choices of numbers in the above definition of \(\mathcal {R}_{\alpha \beta \gamma }\) can be found in Fig. 9. Also see [15].
Remark 9.1
The \(\hbox {KdV}_K\) reduction \(q_i = -p_i\) yields
which determines a simpler solution of the tetrahedron equation. \(\square \)
10 A generalization of the vector KP R-matrix and a solution of the functional tetrahedron equation
The vector KdV R-matrix (5.7) is obtained from the one-parameter R-matrix (see, e.g., [15, 20] for a similar R-matrix)
by setting \(x = (p_1-p_2)/(p_1+p_2)\). The local Yang–Baxter equation
where indices \(\alpha \beta \) indicate on which components of a threefold direct sum R acts, determines the map \((x,y,z) \mapsto (X,Y,Z)\) given by
A similar map appeared in [15, 20]. A general argument (cf., e.g., [9] and references cited there) implies that
solves the (functional) tetrahedron equation (9.1), where a “product” of \(\mathcal {R}\)’s now has to be interpreted as composition of maps. This tetrahedron map is involutive. Setting \(x = (p_1-p_2)/(p_1+p_2)\), \(y= (p_1-p_3)/(p_1+p_3)\) and \(z= (p_2-p_3)/(p_2+p_3)\), it becomes the identity.
Correspondingly, the vector KP R-matrix (5.6) is obtained from the more general two-parameter R-matrix
by setting \(x = (p_1-p_2)/(p_1-q_2)\) and \(y = (p_1-q_1)/(p_1-q_2)\). The local Yang–Baxter equation
determines the map \((x,y;z,u;v,w) \mapsto (X,Y;Z,U;V,W)\), where
with
Then
solves the functional (i.e., set-theoretical) tetrahedron equation.
11 Conclusions
In this work, we explored “pure” soliton solutions of matrix KP equations in a tropical limit. In case of the reduction to matrix KdV, this consists of a planar graph in (two-dimensional) space-time, with polarizations assigned to its edges. Given initial polarizations, the evolution of them along the graph is ruled by a Yang–Baxter map. For the vector KdV equation, this is a linear map, hence an R matrix. The classical scattering process of matrix KdV solitons resembles in the tropical limit the scattering of point particles in a 2-dimensional integrable quantum field theory, which is characterized by a scattering matrix that solves the (quantum) Yang–Baxter equation.
We have shown that all this holds more generally for \(\hbox {KP}_K\), where the tropical limit at a fixed time t is given by a graph in the xy plane, with polarizations attached to the soliton lines. Moreover, the vector KP case provides us with a realization of the “classical straight-string model” considered in [15]. It should be noticed, however, that KP line solitons in the tropical limit are not, in general, straight because of the appearance of (phase) shifts.
As a side product of our explorations of the tropical limit of pure vector KP solitons, we derived apparently new solutions of the tetrahedron (Zamolodchikov) equation. Whether these solutions are relevant, e.g., for the construction of solvable models of statistical mechanics in three dimensions, has still to be seen.
Another subclass of soliton solutions of the vector KP equation consists of those, for which the support at fixed time is a rooted and generically binary tree in the tropical limit. For the scalar KP equation, this has been extensively explored in [6, 7]. Instead of the Yang–Baxter equation, the pentagon equation (see [9] and references therein) now plays a role in governing corresponding vector solitons. This is treated in a separate work [10].
Notes
More generally, the following holds for any constant\(\phi _0\). But adding to \(\phi \) a constant matrix is an obvious symmetry of the \(\hbox {pKP}_K\) equation.
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Open access funding provided by Max Planck Society. A.D. thanks V. Papageorgiou for a very helpful discussion.
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Dimakis, A., Müller-Hoissen, F. Matrix KP: tropical limit and Yang–Baxter maps. Lett Math Phys 109, 799–827 (2019). https://doi.org/10.1007/s11005-018-1127-3
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DOI: https://doi.org/10.1007/s11005-018-1127-3