Toda-Heisenberg chain: interacting sigma-fields in two dimensions

where σ is a three-component vector of unit length, (σ, σ) = 1.

(1.2)

and braces denote the standard scalar product. The energy of our system is given by

with nearest-neighbour interaction

of the Heisenberg type:

dx dy F (σ n , σ p ) .

(1.5)

The models of this type can appear, for example, in the studies of the lamellar (graphitelike) magnetics when the spin interaction inside one layer can be described in the framework of the Landau-Lifshitz theory with effective Heisenberg interaction between adjacent layers. The stationary structures of our system are governed by the (2+1)-dimensional equation

In what follows we use a function F which is peculiar to integrable nonlinear mathematics (see e.g. [1,2]),

(1.7)

The resulting equations are given by

where

.

(1.9)

The factor g2 can be eliminated by rescaling the coordinates, so we take g = 4 (1.10) and write the central equation of our study as In the following sections, after re-parametrization of (1.11), we split it in Sec. 2 into a first-order system, bilinearize it (Sec. 3) and derive the dark-soliton solutions (Sec. 4).

Using the vector-matrix correspondence

where σ j (j = 1, 2, 3) are the Pauli matrices and introducing complex variables

one can rewrite Eq. (1.11) as

In what follows we use the parametrization of the vectors σ n based on the presentation of the matrices S n in the form

Using the invariance of this representation with respect to transformations Ψ n → D n Ψ n with arbitrary diagonal matrices D n one can choose

which leads to

Calculating ∂ ∂ S n and f np ,

one comes to the following system of equations:

where

)

and

The crucial step of our proceeding is the following ansatz : we split the above system into two first-order ones,

and To summarize, one can obtain a large number of solutions of (2.4) by solving the system

(2.20)

Before proceed further, we would like to give some comments on this system. After introducing new variables, and

and can be identified with the (X 1 , Y 1 ) equations (with a(u, v) = (u -v) 2 ) from the list of the paper by Adler and Shabat [3].

At the same time both B n and C n solve the (2+1)-dimensional version of the Ruijsenaars-Toda lattice [4,5]

Note that equations (2.26) are different from (and complementary to) the Ruijsenaars-Toda lattice (R 1 ) that appears in a natural way in the framework of [3]. Finally, calculating from Eqs. (2.15), (2.16) derivatives of the functions f n defined by

one can demonstrate that these functions satisfy

(2.28)

Thus one can see the relationship of the model discussed in this paper with the famous two-dimensional Toda lattice.

To bilinearize Eqs. (2.19), (2.20) we introduce ρn , τn , τn and σn by

and another set of tau-functions by

and

where α and ᾱ are constants, D and D are the Hirota’s bilinear differential operators, and

where β, γ

and substituting the above formulae into (3.2) and (3.3) one can obtain

and where

Thus, to finish solution of our problem one has impose the condition ). An important question that arises now is the question about compatibility of this system. We do not present here an explicit proof of the fact that Eqs. (3.2)-(3.5) are compatible because (i) we present (in the next section) their explicit solutions and (ii) show their relation to a well-known nonlinear compatible system -the Ablowitz-Ladik hierarchy (ALH) [6]. To do the latter let us consider the matrix

and inverse one can obtain

and

Inspecting (3.20)-(3.24) one can conclude, after eliminating the unnecessary constants, introducing ) and to make the following formulae more readable it seems useful to introduce instead of the triplet ρ n , τ n and σ n an infinite set of tau-functions τ m n ,

In new terms equations (3.2)-(3.5) become

for m = 0, 1 and

for m = 0. These equations are a part of the generalized ALH [7] and can be solved without imposing restrictions on m, for -∞ < m < ∞.

4. Dark solitons.

Here we would like to present some basic formulae describing the dark-soliton solutions of the ALH that we then use to obtain solutions of our problem. The dark solitons for the AL equations were obtained in [8] using the inverse scattering method. In [9] these solutions were derived, using purely algebraic method based on the Faylike identities for the determinants of some special matrices. Here we use notation slightly different from one of [9], which makes the following formulae more simple and clear.

The key objects behind the dark-soliton solutions of the ALH are the determinants

with matrices A satisfying

Here I is the N × N unit matrix, L and R are constant diagonal matrices, In what follows we use ‘shifted’ determinants

where

with

An important property of these determinants, that we repeatedly use below, is the Fay’s identity

which can be proved directly.

Using the limit procedure one can

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