Characters of Explicit Solutions for a Semidiscrete Integrable Coupled Equation
A semidiscrete integrable coupled system is obtained by embedding a free function into the discrete zero-curvature equation. Then, explicit solutions of the first two nontrivial equations in this system are derived directly by the Darboux transformation method. Finally, in order to compare the solutions before and after coupling intuitively, their structure figures are presented and analyzed.
Integrable coupled equations have attracted more attention in soliton theory in recent years. For a given integrable system, we can construct a nontrivial system of differential equations which is still integrable and includes the original integrable system as a subsystem . It is interesting to find integrable coupled systems for a given integrable equation. Hirota and Satsuma in 1981 introduced a coupled KdV system . Fuchssteiner in 1982 proposed the important question: how should completely systems interact without losing complete integrability ? Then, the method for constructing integrable coupling systems by perturbation was first proposed by Ma and Fuchssteiner . Later, the method has been developed. So far, it mainly includes perturbations, enlarging spectral problems [5, 6], creating new loop algebras , and multi-integrable couplings. With the development of integrable coupling theory, it is verified that integrable coupled equations are usually used for describing phenomena related to dark and antidark solitons. Such as the integrable coupled generalized nonlinear Schrödinger equations can exhibit N-bright-bright and N-dark-dark soliton solutions .
Obtaining explicit solutions for integrable equations is the main mission in nonlinear science research. Many methods have been developed to solve integrable equations, such as Darboux transformation [9–15], inverse scattering transformation , algebra-geometric approach , Lie symmetry method , and Hirota bilinear method . Darboux transformation is a useful tool for solving integrable equations. It can obtain its nontrivial solutions in accordance with an arbitrary seed solution of the integrable equations. Solving integrable coupled equations by the Darboux transformation method is a meaningful investigation. Explicit solutions of an integrable coupled system of Merola-Ragnisco-Tu lattice equation (24) are investigated, and explicit solutions of a new discrete integrable soliton hierarchy with Lax pair  are discussed by the Darboux transformation.
In this paper, our main consideration is the following semidiscrete integrable coupled equations: where and are the potentials. The spectral problems deriving the semidiscrete integrable coupled system are an extension of a spectral problem introduced by Sun et al. . Another extension of the spectral problems in  was investigated by Xue et al. . They also constructed infinitely many conversation laws and Darboux transformations for the first nonlinear integrable equations. In this paper, we will concentrate on investigating explicit solutions of Equation (1) by means of the Darboux transformation method based on its Lax pair.
The outline of this paper is as follows. In Section 2, we will derive Equation (1) by means of the discrete zero-curvature equations and construct the Darboux transformation for Equation (1) based on its spectral problems. In Section 3, we will obtain the explicit solutions of Equation (1) and discuss the properties of solutions by means of different figures. In Section 4, some conclusions will be given.
2. A New Integrable Coupled Equation and Its Darboux Transformation
2.1. Constructing for a New Integrable Coupled Equation
In order to obtain Equation (1), we consider the following discrete spectral problem: and its auxiliary problem where is independent of and is a shift operator defined by .
Then, we embed a free function into the following modification term :
Let . Solving the stationary discrete zero-curvature equations and the discrete zero-curvature equations , we can obtain the following integrable coupled hierarchy:
2.2. Darboux Transformation
In this section, we will investigate the Darboux transformation of Equation (1).
The spectral problems of Equation (1) are presented by where
Firstly, we choose a proper Darboux matrix:
The Darboux transformation can transform into , i.e.,
And the potential functions and have the same form as and , respectively. Then, we need to define a solution matrix ; it can be represented as
If we assume that the terms both and are two linear independent solutions of the Lax pair of Equation (1), we can obtain by means of the transformation
So we have an algebraic system:
Proposition 1. The form of the matrix is and the transformation from the old potentials to new ones is given by
Proof. Let and where all the expressions are the functions with respect to and . It is easy to verify that the terms and are the roots of except , and . We can also prove that , the terms are ninth-order polynomials with respect to , and the terms and are tenth-order polynomials with respect to . The equation can be written as , with By comparing the coefficients in , we find From the above equations, we can see . The proof is completed.
Proof. where all the expressions are the functions with respect to and . It is obvious that the terms , and are tenth-order polynomials with respect to , the terms , and are ninth-order polynomials with respect to , the term is eighth-order polynomials with respect to , and the terms , and are all zero. In addition, the terms and are the roots of the . So, we have with Equation (25) can be rewritten as By comparing the coefficients in Equation (27), we obtain It is obvious that . The proof is completed.
3. Explicit Solutions
3.1. Explicit Solutions for Equation (1)
Then, based on Equation (13), we can obtain
The integrable equations investigated in  are presented as follows:
In order to compare the differences of the explicit solutions between integrable coupled Equation (1) and integrable Equation (32), we first investigate the three-dimensional structure and the evolution properties of explicit solutions Equation (33) with two sets of parameters, , and and , , , and in Figures 1 and 2. Then, we plot the three-dimensional structures, density figures, the evolution properties of , and with parameters , , and , and with parameters , and in Figures 3–5 (The expression for is so complicated that Maple cannot calculate its figure when the parameters contain a complex number. The specific expression of is in the appendix.).
(a) Density plot of
(b) Density plot of
(c) Density plot of
(d) Density plot of
(a) The evolution of
(b) The evolution of
(c) The evolution of
(d) The evolution of
From Figures 1 and 2, it can be observed that the solitary waves move from right to left whether the parameter contains complex numbers or not. From the evolutions of solutions in Figure 1, we see that these one-soliton solutions are not stable. From Figure 2, we observed that and have the kink-shaped structures when we take the appropriate parameters. From Figure 5, it is easy to see that the solitary waves of , , and also move from right to left, and they are also not stable. From Figures 1–5, we find that the shapes, amplitudes, and wavelengths of solutions have a big difference before and after coupling, but the directions of solitary wave solution propagation have not changed.
In this paper, we have constructed a coupled Lax pair by enlarging the Lax pair and derived new semidiscrete integrable coupled equations which include the original integrable equations as subequations. Next, we have found a suitable Darboux matrix and obtained the explicit solutions of Equation (1) according to the seed solution by means of the Darboux transformation method. Then, we plot three-dimensional structure figures, the evolution properties of solutions before and after coupling, and the density plot of solutions after coupling. Finally, we analyze the properties of solutions before and after coupling. All the results in this paper may be helpful in understanding some physical phenomena.