Abstract
Using a suitable gauge transformation matrix, we present a -fold Darboux transformation for a Vakhnenko lattice system. This transformation preserves the form of Lax pair of the Vakhnenko lattice system. Applying the obtained Darboux transformation, we arrive at an exact solution of the Vakhnenko lattice system.
1. Introduction
Since the beginning of this century, the integrable lattice systems (or lattice soliton systems) have received considerable attention. Many important integrable lattice systems have been studied from the perspective of Mathematics and Physics, for instance, the Ablowitz-Ladik lattice [1], the Toda lattice [2], and the relativistic Toda lattice [3], [4–11].
In the soliton theory, the Darboux transformation is a very effective method for solving soliton equations [12, 13]. Later, it also applies to solving integrable lattice equations [7–9, 14, 15]. In reference [10], a Vakhnenko lattice system is introduced:where , , is a discrete variable, , and is a continuous variable, . Equation (1) can be rewritten as a discrete zero curvature equationof a discrete spatial spectral problemand a corresponding continuous time evolution equation
Here, for a lattice function , the shift operator and the inverse of are defined as follows.
In equations (3) and (4), is the eigenfunction vector, is a potential vector, (1) is a very meaningful lattice system, and many important lattice systems can be reduced from it, such as the nonlinear self-dual network equation, the two coupled discrete nonlinear Schrödinger equation, and the relativistic Volterra lattice [9]. In reference [9], the authors discussed the Darboux transformation of (1), but their results are incorrect [11]. In reference [10], although the author gave some properties of the Darboux transformation of (1), but his approach is different from ours. In reference [11], we derived a 1-fold Darboux transformation of (1). In addition, let , and the equation (1) is reduced to a three-component differential-difference system.
In reference [14], its N-fold Darboux transformation is presented. Furthermore, if we set , equation (1) becomes the nonlinear self-dual network equation.
In reference [15], the author derived its N-fold Darboux transformation. In this letter, for arbitrary positive integer N, we will present a -fold Darboux transformation for the Vakhnenko lattice system (1). Finally, an exact solution of (1) is derived.
2. N-Fold Darboux Transformation
For any positive integer , we introduce the following matrix:
Here, , , are undetermined constants.
Next, we consider the gauge transformation [7, 8].
By the transformation (9), the Lax pairs (3) and (4) become
Equations (10) and (11) constitute a new Lax pair. Let us denoteare two real linear independent solutions of equations (3) and (4), and are distinct eigenvalues of spectral problem (1).
Proposition 1. The matrix has the same form as in equation (3), and the transformation formula is presented byNamely,In the above matrix, are determined by (15), and they are all independent . Obviously, (15) transform the old potentials of (1) into the new potentials of (9).
Proof. We consider the following linear system:Here,where are suitably chosen, such that all the determinants of coefficients for the equation and (17) are nonzero. By solving the linear system (17), we get , .
From equations (3) and (18), we haveFor convenience, we setwhereThen, we obtain the following equation:where is the adjoint matrix of andIn the light of (17), we can getAnalyzing the coefficients of powers of λ in (22), we obtain thatHere,where are all independent of . By comparing the coefficients of in (25), we haveThe proposition is proved.
Proposition 2. Under the transformation (15), the matrix has the same form as in equation (2). In other words,
Proof. We considerwhereOwing to and paying attention to the coefficients of powers of in (29), we findIn equation (31),where are all independent of spectral parameter . Comparing the coefficients of in equation (31), we arrive atThe proposition is proved.
In summary, we get the following theorem.
Theorem 1. The equations (9) and (15) constitute a Darboux transformation of (1), that is, from an old solution of (1), through transformation (15), a new solution of (1) is derived.
3. An Exact Solution
In what follows, we will derive a solution of equation (1) by the Darboux transformation (15). For simplicity, we consider the case of .
First, we select the seed solution of the lattice system (1), namely, the simple special solution, . Substituting this solution into the corresponding Lax pair, we get
Solving the above two equations, we get two real linear independent solutions:
Then, we have
Here, are four different arbitrary constants. When ,
According to Proposition 1, we can get the following Darboux transformation:
Here,
Thus, we can obtain a solution of equation (1) as follows:
In equation (40),
Here,
4. Conclusion
In this work, by means of a gauge transformation of Lax pair, we established a -fold Darboux transformation for a Vakhnenko lattice system. Under this transformation, the structure of the Lax pair remains unchanged. Finally, as an application of this transformation, an exact solution of the Vakhnenko lattice system (1) is given. Starting from the exact solution (40), we apply the Darboux transformation (38) once again; then, another new solution of equation (1) is derived. This process can be performed continually. So, we can get many exact solutions for the lattice system (1).
Data Availability
The data used to support the findings of this study are available from the corresponding author reasonable request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Authors’ Contributions
Ning Zhang completed computing. Xi-Xiang Xu proposed the problem, drafted the manuscript, read, and approved the final manuscript.
Acknowledgments
This work was supported by the Natural Science Foundation Project of China (11805114).