The periodic solutions of the discrete modified KdV equation with a self-consistent source
Introduction
In 1967, Gardner et al. [1] proposed the method of the inverse scattering for the Schrödinger equation on the line as a method for solving the Cauchy problem for the Korteweg–de Vries (KdV) equation
Shortly thereafter, Lax [2] pointed out the general character of the inverse scattering method. A few years later, Zakharov and Shabat [3] managed to solve another important nonlinear evolution equation, the so-called nonlinear Schrödinger equation, using a nontrivial extension of the methods used in [1], [2]. Thus, the way for constructing some other classes of equations solvable by the inverse scattering method was opened. A detailed exposition of the relationship between the inverse scattering theory and nonlinear equations of mathematical physics can be found, for example, in the monographs [4], [5], [6], [7].
In attempts to identify a wider class of integrable systems, an important role has been played by the squares of eigenfunctions of Sturm–Liouville eigenvalue problems; this was revealed in [8]. Newell [9] showed that the squares of eigenfunctions rather than the eigenfunctions themselves are essential in integrating, by the inverse scattering method, nonlinear equations related to the Sturm–Liouville equation. This fact was rigorously proved by Calogero and Degasperis [10].
Since late 1980s and early 1990s, integrable Hamiltonian ODEs as well as integrable symplectic maps were constructed by taking “restricted flows” or “Bargmann-constrained flows” of integrable nonlinear evolution equations, both in the continuous and discrete cases. These finite dimensional systems have been readily recognized to be stationary flows of nonlinear evolution equations with self-consistent sources. As noted in [11], some generalizations of the Lax equations in the theory of integrable equations have been proposed by Melnikov [12]. Later they were also derived by Zakharov [13]. The general form of these equations becomes the generalization of the standard Lax form obtained by adding operator C, namelywhere L and A are linear differential operators also depending on the time parameter and acting on a fixed Hilbert space, and C is the sum of differential operators with coefficients depending on solutions of the spectral problem for the operator L. These equations are usually called equations with self-consistent sources.
Soliton equations with self-consistent sources have received much attention in the recent literature. Physically, the sources appear in solitary waves with non-constant velocity and lead to a variety of dynamics of physical models. For applications, such systems are usually utilized to model interactions among different solitary waves and they are relevant in significant problems in various areas such as hydrodynamics, solid state physics, and plasma physics [14], [15], [16], [17], [18]. Various techniques have been used to construct their solutions, such as the inverse scattering method [15], [16], [19], [20], the method of Darboux transformations [21], [22], [23], [24], and the Hirota bilinear method [25], [26], [27].
The periodic Toda lattice was considered in the works [28], [29], [30], [31], [32], [33]. In the works [34], [35] the inverse spectral problem is applied for integration of the periodic Toda lattice with self-consistent source.
The Ablowitz–Ladik (AL) lattice is an important difference-differential system which was first introduced by Ablowitz and Ladik as the discrete counterpart of the AKNS system [36], [37]. The AL lattice and its reductions, such as the discrete nonlinear Schrödinger equation (DNLS) and the discrete modified KdV (dmKdV) equation, have been studied intensively from both mathematical and physical points of view. Several methods have been used such as the Inverse Scattering Transform [37] and the Backlund transformations [38] to construct their solutions. The Darboux transformations and explicit solutions to the AL lattice with self-consistent sources (ALESCS) were studied in [39].
In the work [40] the algebro-geometric initial value problem for the Ablowitz–Ladik hierarchy with complex-valued initial data is considered and proof of unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure is discussed.
The discrete analog of the periodic variant of the inverse scattering technique to construct exact periodic solutions of the dmKdV equation is used in [41].
This work is devoted to the application of inverse spectral problem for integration of the periodic dmKdV equation with self-consistent source. The effective method of solution of the inverse spectral problem for the corresponding discrete spectral problem is presented.
In this work, the discrete modified Korteweg-de Vries (dmKdV) equation with self-consistent sourcein the class of N periodical functions with the initial conditionsis studied. Here n ∈ Z are given N-periodical sequence, and the numbers zk, are solutions of the equation where . By cn(z, t), n ∈ Z and sn(z, t), n ∈ Z we denote the column vector solutions of system of equationsunder the initial conditions and .
The function sequencesare unknown vector-functions, besides where are the Floquet-Bloch solutions for the system of Eq. (3) which is normalized by conditions and defined by Remark Since the numbers zk and are solutions of equation it is easy to see thatThe main aim of this work is to provide a representation for the solutions of the problem (1)–(3). The Lax pair associated with Eq. (1) is written asHere Hn and Fn are matrices such asandwhereIt is easy to see that the consistency between the equations of system (4) is equivalent to Eq. (1) and can be written asBy direct calculation we can find thatSo,
Section snippets
The discretized eigenvalue problem
In this section we give basic information about the theory of the direct spectral problem for the discrete linear Ablowitz–Ladik equation.
We consider the following Ablowitz–Ladik’s systemwith spectral parameter z. Here qn ≠ ± 1 and n ∈ Z.
Let cn(z), n ∈ Z and sn(z), n ∈ Z be the solutions of Eq. (7) under the initial conditions and .
We will introduce polynomials Tn(λ), Sn(λ), Pn(λ) and Un(λ) in degree of n by the
The algorithm for solving the inverse spectral problem
The set of the numbers and the spectral parameters ξj, is called spectral data of Eq. (3). Finding qn, n ∈ Z from the spectral data is called the inverse spectral problem for the Eq. (3). For solving the spectral problem we need an algorithm for finding the spectral data in the kth step through the spectral data in the th. To this and we deduce relations between polynomials and and polynomials QN,k(λ) and . First
Evolution of spectral parameters
Theorem 4.1 If the functions qn(t), are solutions of the problem (1)–(3), then the spectrum of the periodic and antiperiodic problems for Eq. (3) is independent of the parameter t and the spectral parameters ξj(t), satisfy the following system of equations Proof We can rewrite the monodromy matrix MN in the following formThe derivative of MN with respect to t is
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