Solitons of some nonlinear sigma-like models

We present a set of differential identities for some class of matrices. These identities are used to derive N-soliton solutions for the Pohlmeyer nonlinear sigma-model, two-dimensional self-dual Yang-Mills equations and some modification of the vector Calapso equation.


Introduction.
This paper is a continuation of a cycle work devoted to the derivation of the soliton solutions for various integrable models (see [12,17,13,14,15,16] and references therein). In all these studies we exploit the already known fact that soliton solutions of almost all integrable equations possess similar structure which can be clearly expressed in terms of some class of matrices [12,17].
In this work we consider the following three models. The first one is the Pohlmeyer nonlinear sigma-model [7,9], described by the action where ∂ ζ stands for ∂/∂ζ, which, depending on whether the variables are real or complex and on the choice of the involutionū = κu * (with star indicating the complex conjugation and κ 2 = 1) becomes either Getmanov system studied in [4] or the O(3, 1) sigma-model discussed in [10,11]. Also we consider a modification of the vector Calapso equation where ϕ is a complex 4-vector, ϕ ∈ C 4 , / ∂ is a two-dimensional Dirac-type operator and f = f (/ ∂ϕ).
The third equation discussed in this paper, where U is a 2 × 2 complex matrix, can be considered as a two-dimensional reduction of the self-dual Yang-Mills equations [18,8,6]. As in the works cited above, we do not address the questions of integrability and do not employ the inverse scattering transform. Instead, we use some kind of the so-called 'direct' method. We introduce some functions constructed of the soliton matrices studied in [17] and present in section 2 a set of algebraic and differential identities for these functions (we will use the term 'auxiliary system' for this set). Then we demonstrate in section 3, by some elementary calculations, that functions satisfying this auxiliary system can be used to construct solutions for the equations that we study in this paper. In section 4 we focus on the questions related to the complex conjugation and demonstrate that to perform the corresponding reductions in the framework of our direct approach is much easier than in the framework of the inverse scattering transform or the algebro-geometric approach. Finally, in the last section we give a few comments about the obtained results.

Auxiliary system.
We start with the so-called 'almost-intertwining' matrices [5] that satisfy the 'rank one condition' [2,3],L Here, L andL are diagonal constant N × N matrices, |α and |ᾱ are constant N -component columns, a| and ā| are N -component rows that depend on the coordinates describing the model. It should be noted that throughout this paper the overbar does not mean the complex conjugation (which will be indicated by the * -symbol).
The ξ-and η-dependence of the matrices A andĀ that we use in this study is defined by where C andC are constant N ×N matrices. Note that C andC are not arbitrary: their structure is determined by (2.1). Now, our task is to calculate derivatives of various combinations of the matrices A andĀ, matrices G andḠ defined by rows a|, ā| and columns |α , |ᾱ . In particular, we are going to derive a closed set of differential identities involving the eight functions and Using straightforward calculations one can obtain the following identities involving the ∂ ξderivatives. (2.4) and (2.5) satisfy the following set of equations In a similar way, one can derive the set of ∂ η -identities.
Proposition 2.2. Functions u,ū, v, v,v, w 3 , w 4 defined in (2.4) and (2.5) satisfy the following set of equations We do not present here a proof of all of these identities. In appendix A a reader can find examples of how to obtain some of them, while the rest can be derived in an analogous way.
An immediate consequence of these results is that the function I defined as I = uū + vv is constant: ∂ ξ I = ∂ η I = 0. More careful analysis leads to the identity uū + vv = 1 (2.9) which will be often used in what follows. It turns out that a derivation of this simple identity is the most cumbersome part of the calculations of this paper. We present a proof of (2.9) in appendix B. System (2.7)-(2.9) is not new. It is closely related to the Ablowitz-Ladik hierarchy [1], which is not surprising because, as is shown in [17], the bright solitons of the Ablowitz-Ladik hierarchy are built of matrices A andĀ (2.1) and have the structure of functions defined in (2.4). However, we do not discuss these questions here and consider (2.7)-(2.9) as a closed set of identities which is used in what follows to construct explicit solutions for the equations which are subject of this paper.

N-soliton solutions.
In this section we construct N-soliton solutions for the equations listed in the introduction. Identities (2.7)-(2.9) give us a possibility to do this with very little effort.

Sigma-model.
Starting from equations (2.7) and (2.8), it is easy to derive the following identities involving second derivatives of the functions u andū Noting that vv = 1 − uū, we can rewrite equations (3.1) and (3.2) as These equations are nothing but the Euler equations for the action with the Lagrangian given by In a similar way, one can derive from (2.7) and (2.8) the identities rewrite them as and note that they correspond to the Lagrangian These calculations can be summarized as follows.

4) satisfy the Euler equations for the Lagrangian
Thus, functions defined in (2.4) provide solutions for the field equations for the Pohlmeyer nonlinear sigma-model.

Calapso equation.
The elementary consequence of the equations (2.7) and (2.8) is the fact that the four-vector ϕ defined by obeys the identity Moreover, one can express the function f in terms of w 1 , w 2 , w 3 and w 4 by noting that which, together with (2.9), leads to This means that (3.17) can be presented as a closed equation for the vector ϕ . After introducing the Dirac operator by noting that ∂ ξη = −/ ∂ 2 and rewriting the identity (3.20) in terms of ϕ as one can present this result as Proposition 3.2. Vector ϕ given by (3.16) where functions w 1 , w 2 , w 3 and w 4 are defined in (2.5), is a solution for the equations which describes some vector variant of the sigma model discussed above.

2D Self-dual Yang-Mills-like equations.
To derive the soliton solutions for the two-dimensional self-dual SU (2) Yang-Mills-like equations consider the matrix which, due to (2.9), belongs to SL(2, C), det U = 1. Equations (2.7) and (2.8) imply After differentiating these identities once more and using, again, (2.7) and (2.8), one can arrive, after some simple calculations, at and It easy to demonstrate that the right-hand side of the last equation can be rewritten as To summarize, we have derived the following result.

4) satisfies the two-dimensional self-dual Yang-Mills-like equations
It is easy to see that if one starts from the system (2.7)-(2.9), then to derive soliton solutions for equations (3.15), (3.24) with (3.25) or (3.34) becomes a rather easy task.

Solitons with involution.
Till now, we have not raised the questions related to the complex or Hermitian conjugation: the functions u andū, v andv as well as w 1 , w 2 , w 3 and w 4 have been treated as independent. However, in physical applications of the models discussed in this paper these questions are very important. Thus, in this section we study the properties of the described in the previous section solutions from this viewpoint.
Our first problem is to determine conditions which ensure the following identity: where * stands for the complex conjugation. Analyzing the compatibility of equations (2.1) and equations (2.2), which determine the ξ-and η-dependence, with the complex conjugation on can distinguish two important cases L = (L * ) ±1 . In what follows, we consider these cases separately and see how the involution modifies equations (3.15), (3.24) with (3.25), (3.34) and their solitons.

'Minkowski' case.
In this case the diagonal matrices L andL, the rows a| and ā| and the columns |α and |ᾱ are related bȳ while the variables ξ and η should satisfy conditions ξ = −ξ * and η = −η * . After introducing real variables t and x by the row a| and the matrix A can be presented as Here c| is an arbitrary constant row, C is the constant matrix given by where c j , α j and L j are the components of c|, |α and L, which play the role of the constant parameters of the N-soliton solutions 1 , and The restrictions (4.2) lead to the following relations between the functions which are defined in (2.4) and (2.5): Now we can reformulate some of the results presented in the previous section.
and summation over µ is understood.
Considering the Calapso equation (proposition 3.2), it should be noted that due to the symmetry (4.8) we can rewrite it as an equation for the C 2 vectors ψ = 1 √ 2 (w 1 , w 3 ) T . After the redefinition of the Dirac operator, one can easily verify that (/ ∂ψ) † (/ ∂ψ) = 4|u| 2 |v| 2 (4.12) which leads to the following result.
where functions w 1 and w 3 are defined in (2.5)

'Euclidean' case.
In this case the diagonal matrices L andL, the rows a| and ā| and the columns |α and |ᾱ are related bȳ and η = ξ * . After introducing real variables x and y by the row a| and the matrix A can be presented as Here c| is an arbitrary constant row, C is the constant matrix given by where c j , α j and L j are defined in (4.6) and  where ∇ is the gradient operator, ∇ = (∂ x , ∂ y ) T .
for the equation First, we want to give a comment about the interrelations between the models discussed in this paper. As one can easily see, all models considered here are closely related to the auxiliary system (2.7), (2.8). At the same time, it has been shown in [10] that the sigma-model from section 3.1 can be described in terms of the Ablowitz-Ladik hierarchy. This indicates that both vector Calapso equation (3.24), (3.25) and matrix Yang-Mills-type equation (3.34) can also be 'embedded' into the Ablowitz-Ladik hierarchy which is usually associated with the evolutionary equations like the discrete nonlinear Schrödinger or modified KdV equations.
Next, we would like to point a reader's attention to one of the advantages of the direct approach. If we were trying to derive soliton solutions for the different cases of, say, the general Pohlmeyer sigma-model [7] (the so-called O(4) version of [4] and the O(3, 1)-model of [10]) in the framework of the inverse scattering transform, we would need to elaborate, actually, two distinct versions of the inverse scattering transform, due to different analytic structures of the underlying scattering problems. At the same time, in section 4 we considered both O(4) and O(3, 1) versions on an equal footing: the only difference is the restriction on the constant parameters given by (4.2) and (4.16). Moreover, we have simultaneously obtained both the so-called dark solitons of the model with the Lagrangian (4.9) and the bright solitons of the model (4.10) which, again, needs separate consideration in the framework of the inverse scattering transform.
Finally, we would like to note that three models considered in this paper are far from being the only ones whose solutions can be 'extracted' from the rather simple system (2.7)-(2.9). We hope that the studies presented here can be successfully continued to find other soliton models with possible physical applications.
Consider, for example, the function v. Calculating the derivative of G, and using the derivative of ā|, one can derive Calculations involving the functions u andū are slightly more complicated. Say, to calculate the derivative ∂ ξ u we, first, rewrite (A.1) in an equivalent form, which leads to