Elliptic soliton solutions of the spin non-chiral intermediate long-wave equation

We construct elliptic multi-soliton solutions of the spin non-chiral intermediate long-wave (sncILW) equation with periodic boundary conditions. These solutions are obtained by a spin-pole ansatz including a dynamical background term; we show that this ansatz solves the periodic sncILW equation provided the spins and poles satisfy the elliptic A-type spin Calogero-Moser (sCM) system with certain constraints on the initial conditions. The key to this result is a Bäcklund transformation for the elliptic sCM system which includes a non-trivial dynamical background term. We also present solutions of the sncILW equation on the real line and of the spin Benjamin–Ono equation which generalize previously obtained solutions by allowing for a non-trivial background term.


Introduction
In a recent paper [1], we introduced and solved new soliton equations related to the A-type spin Calogero-Moser (sCM) systems of Gibbons and Hermsen [2] (see also [3]).One of these equations, the spin non-chiral intermediate long wave (sncILW) equation, was shown to have multi-soliton solutions with dynamics described by the hyperbolic sCM system.In this paper, we generalize these solutions to the periodic case.More specifically, we construct periodic solutions of the sncILW equation with dynamics described by the elliptic sCM system [4].This generalization is non-trivial in several regards; in particular, our solutions include a dynamical background term which, as we show, provides a non-trivial generalization even in the hyperbolic limit when the spatial period becomes infinite.We also present corresponding generalizations of known solutions to the spin Benjamin-Ono (sBO) equation introduced in [1] and, in this way, obtain the full correspondence between sCM models and soliton equations conjectured in [1].
A prominent feature of the sncILW equation is its nonlocality, which arises through integral operators (see (1.4)-(1.5)below).While soliton equations with this feature have been studied for a long time (the classical examples are the Benjamin-Ono [5,6] and intermediate long wave [7,8] equations), there has recently been considerable interest in constructing and analyzing novel nonlocal soliton equations; see, for instance, [9,10,11,12,13,14,15,16].In this context, in addition to resolving a conjecture posed in [1], this paper serves to exhibit the sncILW equation with periodic boundary conditions as an interesting system worthy of further study.
Throughout this paper, we denote by ζ(z) and ℘(z) the usual Weierstrass ζ-and ℘-functions of a complex variable z with half-periods (ℓ, iδ), with ℓ > 0 and δ > 0 fixed parameters and i := √ −1 (for the convenience of the reader, we give the definitions of these functions in Appendix A. 1).We find it convenient to use the following variants of ζ(z), and the functions (1.2) see Appendix A.1 for details.The function ζ 1 (z) is 2ℓ-periodic (but not 2iδ-periodic) and the function ζ 2 (z) is 2iδ-periodic (but not 2ℓ-periodic).Note that κ(z) reduces to a constant in the limits ℓ → ∞ and/or δ → ∞ (as can been seen by evaluating κ(z) in (1.2) using the corresponding degenerations of the functions ℘ 2 (z) and ζ 2 (z) presented below in (2.2) and (2.1), respectively).This is one reason why the cases treated in [1] are significantly easier than the elliptic case treated in the present paper.We also note that ζ 2 (z) = ζ 1 (z) + γ 0 z with the constant The constant γ 0 is non-zero only if both ℓ and δ are finite; this is another reason why the elliptic case is more complicated than the cases treated in [1].

Periodic sncILW equation
For d a fixed positive integer, we denote by C d×d the algebra of complex d × d matrices.The periodic sncILW equation describes the time evolution of two C d×d -valued functions U = U(x, t) and V = V(x, t) of x ∈ R and t ∈ R as follows, together with the requirement that both functions are 2ℓ-periodic, U(x + 2ℓ, t) = U(x, t) and V(x + 2ℓ, t) = V(x, t), where [•, •] and {•, •} denote the commutator and anti-commutator of square matrices, respectively, and T and T are integral operators acting on 2ℓ-periodic functions f (x) of x ∈ R as where the dashed integral indicates a principal value prescription and T and T act component-wise on matrix-valued functions.Note that, for d = 1, the sncILW equation reduces to the periodic non-chiral ILW equation introduced and studied by us in [17,18].

Main result
The solutions of the periodic sncILW equation (1.4) that we construct have the form where (1.7) Here, the time-dependent variables M(t) ∈ C d×d , a j (t) ∈ C, and P j (t) ∈ C d×d are such that is time-independent.We refer to M(t) as the background and to a j (t) and P j (t) as pole and spin degrees of freedom, respectively.Our main result is that, by setting (our notation is explained in Section 1.4 below) with |e j (t) and f j (t)| vectors in a d-dimensional complex vector space V and its dual V * , respectively, (1.6)- (1.8) gives an exact solution of the periodic sncILW equation (1.4) provided the following conditions are fulfilled: (i) The dynamical variables {a j , |e j , f j |} N j=1 = {a j (t), |e j (t) , f j (t)|} N j=1 evolve in time according to the following equations, for j = 1, . . ., N .
(ii) The dynamics of the background M = M(t) is given by where P j = P j (t).
(iii) At time t = 0, the following conditions are fulfilled, for j = 1, . . ., N , together with and P † = P. (1.17) (iv) The time t is small enough that the poles neither leave the strip defined in (1.15) nor collide (see Theorem 2.1 for a more precise formulation; as will be discussed, this is a technical condition needed in our proof but which probably can be ignored).
Several remarks are in order.
2. It is important to note that (1.10) and (1.11) are the time evolution equations of the elliptic sCM model [2].(Note that the elliptic sCM model is usually defined with the standard Weierstrass ℘-function ℘(z) instead of ℘ 2 (z); however, this difference is irrelevant since the system of equations (1.10)-(1.11) is invariant under the transformation with P in (1.8) Hermitian, for arbitrary c ∈ R.) 3. We emphasize that the conditions (1.13)-(1.17)are constraints on initial conditions.If (1.13)-(1.17)are fulfilled at time t = 0, then the time evolution equations (1.10)-(1.12)guarantee that (1.13)-(1.17)hold true for all times (this is easy to check for (1.13), (1.16) and (1.17), guaranteed by our assumptions for (1.15), and proved in Proposition 4.1 in Section 4 for (1.14)).
4. While the solutions given above are Hermitian, U = U † and V = V † , we will actually prove a more general result providing non-Hermitian solutions of the periodic sncILW equation and obtain the Hermitian solutions as a corollary; see Theorem 2.1 for the general result.We emphasize the Hermitian solutions here since they are easier to state and probably more interesting in physics applications.

5.
It not obvious but true that the functions U(x, t) and V(x, t) in (1.6)-(1.8)are 2ℓ-periodic as functions of x; to see this, insert ζ 2 (z) = ζ 1 (z) + γ 0 z where ζ 1 (z) is 2ℓ-periodic, and observe that the potentially dangerous term ∝ γ 0 x in (1.7) arising from this insertion vanishes due to the constraint (1.17).
6.We will show in Section 6.2 that the constraints (1.13)-(1.16)can be solved by linear algebra.
7. For ℓ → ∞ and in the special case M = 0, the solutions above reduce to the multi-solition solutions of the sncILW equation on the real line obtained in [1]; it is important to note that one can allow for a non-zero M also in the limit ℓ → ∞ and, in this way, our solutions here generalize the ones in [1] even in the case ℓ → ∞.

8.
The key to our result is a generalization of the Bäcklund transformations of the sCM model in [1] to the elliptic case which is non-trivial since it requires the presence of a non-trivial background M; with this generalization, we obtain a complete correspondence between sCM models and soliton equations, as anticipated in [1] (we discuss this point in more detail in Section 2).We also mention closely related earlier work on Bäcklund transformations of (s)CM systems [19,20].
9. The results in this paper add further support to the conjecture in [1] that the periodic sncILW equation is an integrable system.It is important to note that this would be an elliptic integrable system whose analysis indispensably involves elliptic functions, and this presents challenges not present in the limiting cases ℓ → ∞ and/or δ → ∞ treated in [1].
10.As discussed, the constraint (1.17) is required for the solutions (1.6)-(1.8) to be 2ℓ-periodic and, in this sense, it can be regarded as a balancing condition.We observe that, in the limiting cases ℓ → ∞ and/or δ → ∞, the constraint (1.17) can be ignored; otherwise, the result (with non-zero M) remains true as it stands in these limits (this can be seen by going through our proof of Theorem 2.1 and replacing κ(z) by a constant).

Plan of the paper
In Section 2, we formulate and discuss our main result, Theorem 2.1, which provides, in general, non-Hermitian elliptic soliton solutions to the periodic sncILW equation; corresponding results for the sBO equation are also given.In Section 3, we show that the functions U(x, t) and V(x, t) in (1.6) solve the periodic sncILW equation (1.4) provided that a certain first-order system is satisfied.In Section 4, we establish a new Bäcklund transformation for the elliptic sCM system.Using these results, Theorem 2.1 is then proved in Section 5.In Section 6, we show how to solve the constraints of Theorem 2.1 to generate initial data for our N -soliton solutions, paying particular attention to the one-soliton case.The definitions and functional identities for the Weierstrass elliptic functions that we use are collected in Appendix A. The proofs of two important lemmas stated in Section 4 are deferred to Appendix B.

Notation
We follow [2] and use the Dirac bra-ket notation [21] to write our solutions and relate them to the elliptic sCM system.In particular, we denote vectors in a d-dimensional complex vector space

Results
As mentioned already in the introduction, the periodic sncILW equation is the elliptic case in a general correspondence between sCM models and soliton equations proposed in [1].More specifically, there are four cases in this correspondence which, on the sCM side, can be distinguished by the following special functions which are the well-known two-body interaction potentials in A-type Calogero-Moser systems (see [22] for review).On the soliton side, the functions are the building blocks in the spin-pole ansatz we use to solve the corresponding soliton equations.Note that V (z) = −α ′ (z) in all cases.Moreover, the elliptic case (IV) is most general, and it reduces to the cases I, II and III in the limits (ℓ, δ) → (∞, ∞), δ → ∞ (keeping ℓ finite), and ℓ → ∞ (keeping δ finite), respectively.Nevertheless, Cases I-III are interesting in their own right since, first, they are often sufficient in applications, and second, they are significantly simpler and thus allow for more general results that are not directly obtainable as limits of results for Case IV.
We now give the soliton equations corresponding to Cases I-IV.Cases I and II correspond to the sBO equation given by with a C d×d -valued function U = U(x, t) of x ∈ R and t ∈ R, with H the Hilbert transform; Case I corresponds to the sBO equation on the real line where U(x, t) has suitable decaying conditions at x → ±∞ and and Case II corresponds to the periodic sBO equation, where U(x + 2ℓ, t) = U(x, t), and with the periodic Hilbert transform Case III corresponds to sncILW equation (1.4) on the real line, with functions U(x, t) and V(x, t) of x ∈ R and t ∈ R satisfying suitable decaying conditions at x → ±∞, and with the integral operators T and T defined as (2.6) Finally, Case IV, which is the most general, corresponds to the periodic sncILW equation (1.4) with T and T given by (1.5).Note that, since T → 0 and T → H in the limit δ → ∞ [23], the first equation in (1.4) reduces to the sBO equation (2.3) in this limit.Moreover, T and T in (1.5) reduce to T and T in (2.6) in the limit ℓ → ∞, and H in (2.5) reduces to H in (2.4) in this limit.
For future reference, we summarize the discussion in the present paragraph as follows, sBO equation on the real line (I: rational case) periodic sBO equation (II: trigonometric case) sncILW equation on the real line (III: hyperbolic case) periodic sncILW equation (IV: elliptic case). (2.7) In this section, we present our solutions of the soliton equations in (2.7) in all Cases I-IV.The solutions for the Cases I-III that we present are generalizations of solutions obtained already in [1]; the simplest way to prove these generalizations is to adapt the proofs in [1], as recently done in a thesis by Anton Ottosson [24].This is due to additional constraints appearing in Case IV which prevent a direct derivation of all solutions in Cases I-III as limits of the general solution in Case IV.For this reason, and for clarity, the detailed proofs we present in this paper are restricted to Case IV.

Solutions of the sncILW equation
Throughout this subsection, we consider Cases III and IV in (2.1)-(2.2) and (2.7); in particular, V (z) and α(z) are as in (2.1) and (2.2) for the hyperbolic and elliptic cases.Note that κ(z) = (π/2δ) 2 (a constant!) in Case III, while κ(z) is the non-trivial function in (1.2) in Case IV.
Our general solutions of the sncILW equation (including non-Hermitian ones) are defined in terms of two sets of variables satisfying the time evolution equations of the sCM model: for j = 1, . . ., M (we use the notation in Section 1.4).More specifically, the spin-pole ansatz providing solutions of the sncILW equation (1.4), both in the real-line case (III) and the periodic case (IV), is given in terms of these dynamical variables as follows, where with such that both are time-independent, together with an additional variable M(t) ∈ C d×d describing a nontrivial background.In Case III, N and M are arbitrary and the variable M must be constant, whereas in Case IV, we must have N = M and M is necessarily dynamical.The precise statement is as follows.

16)
with P j = P j (t) and Q j = Q j (t) given by (2.14).(iii) At time t = 0, the following conditions are fulfilled: first, e j |f j = 1 (j = 1, . . ., N ), and third, hold true for all times t ∈ [0, τ ); in Case IV, the same holds true provided that N = M , the conditions in (2.20) hold mod 2ℓ, and the following holds true at time t = 0, We give various remarks related to this result.
1. Theorem 2.1 gives a Hermitian solution, U = U † and V = V † , if and only if N = M and the initial conditions satisfy the following further constraints, which imply Q j = P † j .It is easy to check that the reduction (2.22) is consistent; moreover, if we impose it at time t = 0, it is fulfilled at all times.Thus, in Case IV, we obtain the Hermitian solution of the scnILW equation presented in Section 1.
2. It is important to note the following differences between Cases III and IV: First, in Case III, γ 0 = 0, and therefore U(x, t) = U 0 (x, t) and V(x, t) = V 0 (x, t).However, in Case IV, the functions U 0 (x, t) and V 0 (x, t) given by the spin-pole ansatz (2.13) are related to the solutions U(x, t) and V(x, t) by a time-dependent similarity transformation determined by the total spin P. Second, in Case IV, the background M(t) has non-trivial dynamics, while in Case III, (2.16) simplifies to Ṁ = 0 (since κ(z) reduces to a constant), i.e., the background is constant in time: M(t) = M(0) =: M 0 .Third, in Case IV, we impose the constraint (2.21) on the initial conditions, but this constraint is absent in Case III; as discussed in Section 1.2, this additional constraint in Case IV is required for the 2ℓ-periodicity of U and V (the argument given there straightforwardly generalizes to the more general case here).
3. In Case III, our solutions (2.12) obey the boundary conditions (this follows from lim x→±∞ α(x − a) → ±π/2δ for all a ∈ C); thus, the condition (2.21) is equivalent to U and −V being equal to e iγ 0 (P+Q)t M 0 e −iγ 0 (P+Q)t at x → ±∞; however, there is no need to impose these conditions in Case III.Thus, our solutions suggest that the sncILW equation on the real line is well-defined for the following boundary conditions: where M ±∞ Â are time-independent and, in general, such that 4. The solutions of the sncILW on the real line (Case III) obtained in [1] correspond to the special case M = 0; note that this specialization is only possible in Case III (in Case IV, it is prevented by the non-trivial dynamics (2.16)).[1]; the latter correspond to the special case M = 0 where, again, this specialization is possible in Cases I-III, but not in Case IV.We also remark that, in the elliptic case (IV), a similar Bäcklund transformation for a certain singular limit of the elliptic sCM system, where d = 2 and f j |e j = h j |g j = 0, was recently found by one of the authors in collaboration with Klabbers [25].
6.The Bäcklund transformation, (2.18) with (2.9), (2.11), (2.16) and (2.21), forms an overdetermined system of ordinary equations (ODEs) whose consistency must be established.This was done for the known Bäcklund transformation for the sCM system [2] in Cases I-III in [26] by constructing functions that measure the departure of the Bäcklund transformation from consistency and showing that they obey a system of linear homogeneous ODEs; if the initial data is consistent with the Bäcklund transformation at t = 0, consistency will be preserved at future times, under mild assumptions.While the approach of [26] can be straightforwardly generalized to the Bäcklund transformation in Case IV, leading to essentially the same equations (see [26, Eq. (2.43)]), in this paper we take a more streamlined approach, which allows us to show that the relevant quantities obey a system of linear homogenous ordinary differential equations without deriving its precise form.
7. Theorem 2.1 is stated under the assumption that the poles remain in the strip defined in (2.19) and that no pole collisions occur (see (2.20)).As already mentioned, we believe that these assumptions are unnecessary, and we expect that our soliton solutions can be extended to all times t ∈ R. To support this expectation, we mention the following known results.First, in the sBO case, a Lax pair is known which can be used to prove that the former condition is satisfied for all times [14]; we hope that it is possible to generalize this argument to the sncILW case.Second, for the scalar Benjamin-Ono equation, it is known that pole collisions occur, but they are no problem for the soliton solutions [27].A third reason is recent work by Gérard and Lenzmann on multi-soliton solutions to a nonlocal nonlinear Schrödinger equation [28] (see also [29]) governed by a complexification of the rational CM system; in this work, an explicit example of a two-soliton solution is given where (i) the poles collide and (ii) the solution remains valid during and after the collision.
Clearly, it would be interesting to prove the more general result.

Solutions of the sBO equation
In this section, we consider Cases I and II in (2.1)-(2.2) and (2.7); in particular, V (z) and α(z) are as in (2.1) and (2.2) for the rational and trigonometric cases.
The spin-pole ansatz for solutions of the sBO equation (2.3), both in the real-line and periodic cases, is given by (2.25) with P j (t) and Q j (t) as in (2.14); as in the hyperbolic case, we can consistently assume that the background M(t) = M 0 is time-independent.
Theorem 2.2 (Non-Hermitian solutions of the sBO equation).For fixed N, M ∈ Z ≥0 and 3 From the sncILW equation to a first-order system In this section, we establish conditions under which the ansatz (2.12) solves the periodic sncILW equation.Throughout this section, V (z) = ℘ 2 (z), α(z) = ζ 2 (z), and M = N .
Proof.The proof is facilitated by introducing the notation for C-valued functions F j , G j (j = 1, 2) and the operator interpreted as a linear operator on vector-valued functions, see [17].In the present paper, we use the product • defined in (3.4) also for vectors F, G whose components F j , G j are in C d×d , and we let be the corresponding generalizations of the commutator and anti-commutator, respectively.With this notation, the periodic sncILW equation (1.4) can be written as Using the shorthand notation (a j , |e j , f j |, P j , r j ) := (a j , |e j , f j |, P j , +1) j = 1, . . ., N, (b j−N , |g j−N , h j−N |, Q j−N , −1) j = N + 1, . . ., N , N := 2N, (3.8) we write the ansatz (2.12) as U (x, t) = e iγ 0 (P+Q)t U 0 (x, t)e −iγ 0 (P+Q)t , (3.9) with where It is important to note that, provided (3.2) holds, the quantities P and Q defined in (2.15) and appearing in (3.9) are conserved quantities, using the anti-symmetry of the commutator and the fact that V (z) is an even function (A.5).Thus, assuming (3.2), it follows from (3.7) and (3.9)-(3.10)that U satisfies (3.7) if and only if We compute each term in (3.13) with U 0 given by (3.10).We start with Next, we compute To proceed, we need the identities where The first identity (3.16) can be obtained by differentiating (A.6) with respect to z and setting z = x − a j ± r j iδ/2 while the second identity (3.17) can be obtained by differentiating (A.7) with respect to c, setting a = x, b = a j ± r j iδ/2, and c = a k ± r k iδ/2, and using the periodicity property α(z + 2iδ) = α(z) and the fact that α(z) is an odd function.Inserting (3.16) and (3.17) into (3.15)gives The double sum in the third line and the second double sum in the fourth line vanish by symmetry.Hence, after relabelling summation indices j ↔ k in the double sum in the second line and rearranging, we are left with To compute terms involving T , we need the following lemma.
Lemma 3.1.The operator T defined in (3.5) has the following action on the functions Proof.Using the definitions (3.11) of A ± (z) and recalling that ζ 2 (z) = ζ 1 (z) + γ 0 z with γ 0 given by (1.3), we write By differentiating (3.22) with respect to x, we obtain where ℘ 1 (z) := −ζ ′ 1 (z) is a 2ℓ-periodic, zero-mean function (see Appendix A.1).We compute the action of T on the first and second terms in (3.23) separately.
First, we use the following result [17, Appendix 2.a]: for a 2ℓ-periodic, zero-mean function f (z) analytic in a strip −A < Im(z) < A for some A > δ/2, the vector-valued functions (f (x ∓ iδ/2, −f (x ± iδ/2)) T are eigenfunctions of the T operator with eigenvalues ±i.Applied to the functions f (z) = ℘ 1 (z − a j ), this gives where we have used (3.23) in the second step.and

Bäcklund transformation
Throughout this section, , and M = N .
In this section, we prove that solutions of the elliptic sCM equations of motion (2.8)-(2.11)are, under certain conditions, also solutions of a Bäcklund transformation for the elliptic sCM system.This Bäcklund transformation is given by together with (2.9), (2.11), and (2.16).The precise statement is as follows.
In order to prove Proposition 4.1, we need two lemmas.The first lemma shows that the firstorder system of Proposition 3.1 admits a unique solution under mild assumptions, generalizing a known result in Cases I-III [26].
By differentiating (2.14) with respect to time and inserting (2.9) and (2.11) with (5.1), we find for j = 1, . . ., N .Recalling the definitions of P j and Q j in (2.14) and those of ãj and bj we see that (5.3) is equivalent to (3.2) which thus holds on [0, τ ).
By differentiating the quantities f j |e j and h j |g j with respect to time and inserting (2.9) and (2.11) with (5.1), we find for j = 1, . . ., N .Because (2.17) holds at t = 0 by assumption, (5.4) guarantees it holds on [0, τ ).Thus, by writing P 2 j = |e j f j |e j f j | = f j |e j P j and Q 2 j = |g j h j |g j h j | = h j |g j Q j , we see that (3.3) holds on [0, τ ).
We are now ready to prove Theorem 2.1.

Construction of soliton solutions
In this section, we show how to solve the nonlinear constraints on the initial data in Theorem 2.1.While we specifically reference the constraints in Case IV, the following results can be straightforwardly adapted to Cases III in Theorem 2.1 and Cases I-II in Theorem 2.2.One-soliton solutions are derived in Section 6.1 and a linear algebra problem whose solutions parameterize corresponding multi-soliton solutions is presented in Section 6.2 Given initial conditions M(0) = M 0 , we find the solution M = M 0 , and The constraints (2.17 i.e., f 1,0 | and |e 1,0 are left-and right-eigenvectors of M 0 , respectively.By right-multiplying the first equation in (6.7) by |e 1,0 and left-multiplying the second equation in (6.7) by f 1,0 |, we find Collecting the observations above and using Theorem 2.1, we see that U(x, t) = e 2iγ 0 |e 1,0 f 1,0 |t U 0 (x, t)e −2iγ 0 |e 1,0 f 1,0 |t , V(x, t) = e 2iγ 0 |e 1,0 f 1,0 |t V 0 (x, t)e −2iγ 0 |e 1,0 f 1,0 |t (6.9) with provides a solution of the sncILW equation (1.4) when f 1,0 | and |e 1,0 are left-and righteigenvectors, respectively of M 0 corresponding to the same eigenvalue and normalized to satisfy f 1,0 |e 1,0 = 1 and v 1 is given by (6.8).Remark 6.1.In the generic case where v 1 in (6.8) has a nonzero imaginary part, (6.10) does not provide a traveling wave solution and (2.19) will be violated in finite time, after which Theorem 2.1 does not guarantee (6.10) solves the sncILW equation (1.4).For v 1 to be real, in which case (6.10) provides a traveling wave solution of the sncILW equation (1.4) on [0, ∞), it suffices for M 0 to be Hermitian (in this case f 1,0 | = |e 1,0 † is a possibility but not a requirement unless all eigenvalues of M 0 are simple) and b 1,0 = a * 1,0 .It is interesting to note that in the singular limit of the sncILW equation studied in [25], no such one-soliton, traveling wave solutions exist.

Solution of constraints: Case IV
Throughout this subsection, V (z) = ℘ 2 (z), α(z) = ζ 2 (z), and M = N .For each j = 1, . . ., N , let us identify the vector |e j ∈ V with the vector e j ∈ C d whose components (e j ) µ , µ = 1, . . ., d are the components of |e j with respect to some given basis of V. Next, let us identify the collection of N vectors e j , j = 1, . . ., N , with the single vector e ∈ C N d whose components e j,µ := (e j ) µ are indexed by j = 1, . . ., N and µ = 1, . . ., d.Similarly, let us identify the three collections of vectors { f j |} N j=1 , {|g j } N j=1 , and { h j |} N j=1 with the vectors f = (f j,µ ) ∈ C N d , g = (g j,µ ) ∈ C N d , and h = (h j,µ ) ∈ C M d , respectively.Moreover, consider the matrix representation of M with respect to the same basis of V. We identify M with its vectorization M ∈ C d 2 , i.e., the concatenation of the columns of M.Then, the constraints (2.18) and (2.21) can be written as the (2N + d)d× where We have observed in numerical experiments that the the square matrix in (6.11) is generically rank-deficient.Correspondingly, there are conditions on the vector on the right-hand side of (6.11) for the linear system to be consistent; if f and g are given, these are linear conditions on {v j , w j } N j=1 .If these conditions are satisfied, the solution of (6.11) can be determined uniquely.The constraints (2.17) then lead to an overdetermined linear system of equations for the remaining unknowns in {v j , w j } N j=1 .However, in our numerical experiments, we have found that this system is uniquely solvable.Remark 6.2.Our conventions in this section differ slightly from those in [1,Section 3.1.3],where Hermitian solutions of the sBO equation with M = 0 are considered.There, bolded vectors are always identified with collections of kets, i.e., f is identified with {|f j } N j=1 .We obtain Hermitian solutions of the sncILW equation from (6.11)-(6.12)by setting b j = a * j , f j,µ = e * j,µ , and h j,µ = g * j,µ for j = 1, . . ., N and µ = 1, . . ., d.Note that in this case, the (2N + d)d × (2N + d)d matrix in (6.11) is itself Hermitian.

Solution of constraints: Cases I-III
In this subsection, V (z) and α(z) are as in (2.1) and (2.2) for the rational, trigonometric and hyperbolic cases.The idea of the previous subsection can be adapted to Cases I-III.In these cases, the constraint (2.14) is not present, i.e., we may set C 1 = 0 and C 2 = 0.This yields an underdetermined system in the variables {h, e, M} in (6.11).To generate a (generically) consistent system, we rearrange (6.11) to In Case III, the submatrices appearing in (6.13) are given by (6.12) with α(z) in Case III (2.2).In Cases I-II, the submatrices appearing in (6.13) are given by (6.12) with δ → 0 and α(z) in Cases I-II (2.2).Then, the method described in [1, Section 3.  By the Picard-Lindelöf theorem, the system of equations consisting of (B. holds on [0, τ max ).The overdetermined system of equations for {a j } N j=1 (B.2) will also be satisfied on [0, τ max ) if the difference between the right-hand sides of (B.2) and (B.8) vanish on [0, τ max ).These differences are given by F j | := f j |B j − f j |B j |e j f j | = f j |B j (1 − P j ) (j = 1, . . ., N ), |E j := B j |e j − |e j f j |B j |e j = (1 − P j )B j |e j (j = N + 1, . . ., N ). (B.9) We will show that the time evolution of the quantities { F j |} N j=1 and {|E N +j } N j=1 is determined by a linear, homogeneous (in both { F j |} N j=1 and {|E N +j } N j=1 ) system of ordinary differential equations.The precise form of this system is not needed to establish our result, and so we introduce an equivalence relation ≃ between two expressions that differ only by terms linear in { F j |} N j=1 and {|E N +j } N j=1 with regular coefficients (the regularity of all such coefficients is guaranteed by conditions (i), (ii) in the discussion of maximal solutions above).Thus we only need to show that Ḟj | ≃ 0 (j = 1, . . ., N ), | Ėj ≃ 0 (j = N + 1, . . ., N ).
(B.10)However, in the system of equations given by (2.9), (2.11), (2.16), (2.17), (2.21), and (4.1), the variables {a j , |e j , f j |} N j=1 and {b j , h j |, |g j } N j=1 can be swapped by Hermitian conjugation and consequently, { F j |} N j=1 and {|E N +j } N j=1 can be interchanged using this same symmetry.For this reason, it suffices to show that the first set of equations in (B.10) holds.
We differentiate the first equation in (B.9) with respect to time, which gives (for notational simplicity, we suppress the j-dependence of the quantities The final two lines of (B.25) can be written, using (B.26) and the relations f j |P j = f j | and P j (1 − P j ) = 0, as
19))Then, in Case III, P and Q in (2.15) are both time-independent, and the ansatz (2.12)-(2.15)gives a solution of the sncILW equation on the real line for all times in the interval t ∈ [0, τ ) if τ > 0 is such that (2.19) and 5. It is interesting to note that (2.18), together with (2.9), (2.11), (2.16) and (2.21), is a Bäcklund transformation of the sCM system in the elliptic case (IV); see Section 4 for precise statements.Moreover, in the limits (ℓ, δ) → (∞, ∞), δ → ∞, and ℓ → ∞, this Bäcklund transformation reduces to Bäcklund transformations for the sCM system in Cases I, II and III, respectively; it is important to note that, in Cases I-III, the constraint (2.21) can be omitted; this Bäcklund transformation is a generalization of the Bäcklund transformation of sCM model in Cases I-III obtained in 1.3] can be straightforwardly applied to the generate admissible initial data for Theorems 2.1 and 2.3.towrite (4.1) as ȧj f j | = 2 f j |B j (j = 1, . . ., N ), ȧj |e j = 2B j |e j (j = N + 1, . . ., N ).