New Complex Solutions of the Coupled KdV Equations

(1. School of Mathematics, Inner Mongolia Honder College of Arts and Sciences, Huhhot 010000,China; 2. School of Science, Inner Mongolia Agricultural University, Huhhot 010018, China; 3. School of Mathematical Science, Inner Mongolia Normal University, Huhhot 010022, China)

Abstract: In this paper, new infinite sequence complex solutions of the coupled KdV equations are constructed with the help of function transformation and the second kind of elliptic equation. First of all, according to the function transformation, the coupled KdV equations are changed into the second kind of elliptic equation. Secondly, the new solutions and B¨acklund transformation of the second kind of elliptic equation are applied to search for new infinite sequence complex solutions of the coupled KdV equations. These solutions include new infinite sequence complex solutions composed by Jacobi elliptic function, hyperbolic function and triangular function.

Keywords: The coupled KdV equations; Function transformation; The second kind of elliptic equation; Complex solutions

§1. Introduction

Constructing the multiple soliton solutions of the nonlinear coupled systems is one of important researches in the solitary theory. In [3–7,10–12,23–26], new achievements have been obtained by studying how to search for the exact soliton solutions of the following coupled KdV equations,

The equations (1.1) and (1.2) describe the interaction of two long waves with different dispersion relations, whereα1, α2, α3, β1, β2andβ3are constants.

In [3–7,10–12,23–26], the finite number of single soliton solutions of (1.1) and (1.2) have been given. However, the infinite sequence complex solutions of (1.1) and (1.2) have not been found. This paper will search for the infinite sequence complex solutions of the coupled KdV equations.

First of all, using a function transformation,the coupled KdV equations are transformed into the second kind of elliptic equation. Second, the new solutions and B¨acklund transformation of the second kind of elliptic equation are applied to search for new infinite sequence complex solutions of the coupled KdV equations.

§2. Introduction of the method

2.1. Function transformation and the second kind of elliptic equation

Assume the solutions of the coupled KdV equations (1.1) and (1.2) as

whereλ1, ω1, λ2andω2are constants to be determined and

Whenα1=β1, α3=?α2andβ2=β3=α2,the coupled KdV equations (1.1) and (1.2) will show the second kind of elliptic equations (2.3) and (2.4),

By calculating, the equations (2.3) and (2.4) as follows:

whereξ=λ1x+ω1t, η=λ2x+ω2t, λ1,λ2,ω1andω2are arbitrary constants and not to be zero,andc2are arbitrary constants.

2.2. The relative conclusions of the second kind of elliptic equation

According to the conclusions in [16], the relative conclusions of the second kind of elliptic equation will be presented to seek new infinite sequence complex solutions of the equations (1.1)and (1.2),

Case1.The new solutions of the second kind of elliptic equation.

According to the periodicity of the Jacobi elliptic function, multiple new solutions of the second kind of elliptic equation will be found. Here several kinds of new solutions will be listed.

Whena=1, b=?1?k2andc=k2, equation (2.7) has the following solutions:

Whena=1?k2, b=2k2?1 andc=?k2, equation (2.7) has the following solutions:

Whena=?1+k2, b=2?k2andc=?1, equation (2.7) has the following solutions:

where

Equation (2.7) has the following Riemannθfunction solutions in [16].

Whenequation (2.7) has the following solution:

Whenequation (2.7) has the following solution:

Whenb2?4ac=0, equation (2.7) has the following solutions:

Case2.The B¨acklund transformation of the second kind of elliptic equation.

Ifzn?1(ξ) is the non-constant solution of equation (2.7), then the followingzn(ξ) is also the solution of equation (2.7),

wheredandsare arbitrary constants and not to be zero,aandbare the coefficients of equation(2.7).

Ifzn?1(ξ) is the non-constant solution of equation (2.7), then the followingzn(ξ) is also the solution of equation (2.7),

wherea, bandcare the coefficients of equation (2.7).

§3. The infinite sequence complexion soliton solutions of the coupled KdV equations

The solutions of the second kind of elliptic equations(2.5)and(2.6)are composed by Jacobi elliptic function, the hyperbolic function, the trigonometric function and the rational function.Combination of two forms to construct new infinite sequence complex solutions.

The following superposition formulas are presented to obtain new infinite sequence complex solutions of equation (1.1) and (1.2),

whereFn(ξ) andGn(η) are determined by the following superposition formulas,

Whenki=0 (i=1,2),the solutions of Jacobi elliptic function (3.2) and (3.4) are converted to the solutions of trigonometric function.

Whenki=1 (i=1,2),the solutions of Jacobi elliptic function (3.2) and (3.4) are converted to the solutions of hyperbolic function.

Case1.The new infinite sequence complexion double periodic solutions of equations (1.1) and(1.2).

Subcase1.When;k2=0 or;k1=0,the new infinite sequence complexion double periodic solutions in the peak form composed by elliptic function and trigonometric function.

Subcase2.Whenk1=k2=0,the new infinite sequence complexion double periodic solutions in the peak form composed by two trigonometric functions.

Subcase3.Substituting the solutions determined by the formulas (3.3) and (3.5) into (3.1),then we search for the new infinite sequence complexion double periodic solutions composed by two Riemannθfunctions.

Case2.The new infinite sequence complexion soliton solution and periodic solution of equations(1.1) and (1.2).

Subcase1.When;k2=1 or;k1=1,the new infinite sequence complexion soliton solution and periodic solution in the peak form composed by elliptic function and hyperbolic function.

Subcase2.Whenk1=0;k2=1 ork2=0;k1=1,the new infinite sequence complexion soliton solution and periodic solution in the peak form composed by trigonometric function and hyperbolic function.

Case3.The new infinite sequence complexion double soliton solutions of equations (1.1) and(1.2).

Whenk1=k2=1, the new infinite sequence double soliton solutions in the peak form composed by two elliptic functions.

§4. Conclusion

Auxiliary equation method has acquired many new achievements [1,2,8,9,13–15,17–20,22,27,28] in constructing the exact solutions of the nonlinear evolution equation. The achievements show the following characteristics:

(1) Finite number of the single soliton solutions of single function type.

(2) Finite number of the single soliton solutions of composite function type.

(3) Infinite sequence single soliton solutions of single function type.

(4) Infinite sequence single soliton solutions of composite function type.

Based on [16,21], combining the function transformation, the paper has obtained new achievements by further studying the auxiliary equation method. Existence conditions of new infinite sequence complexion double soliton solutions of equation (1.1) and (1.2) are presented and the new infinite sequence complexion double soliton solutions of (1.1) and (1.2) are found on the base of the function transformation and the relative conclusions of second kind of elliptic equation. These solutions include new infinite sequence complexion double soliton solutions composed by Riemannθfunction,Jacobi elliptic function, hyperbolic function, trigonometric function.