ABSTRACT

In this work, starting from the -expansion method and a variable separation method, a new non-traveling wave general solutions of the (2+1)-dimensional breaking soliton system are derived. By selecting appropriately the arbitrary functions in the solutions, special soliton-structure excitations and evolutions are studied.

1. Introduction

Modern soliton theory is widely applied in many natural sciences [1-4] such as chemistry, biology, mathematics, communication, and particularly in almost all branches of physics like fluid dynamics, plasma physics, field theory, optics, and condensed matter physics, etc. [5-8]. In order to find new exact solutions of nonlinear equations, much methods have been proposed, such as the Lie group method of infinitesimal transformations, the nonclassical Lie group method, the Clarkson and Kruskal direct method (CK) [9,10], the conditional similarity reduction method [11-13] and the improved mapping approach, etc. [14-23].

Recently, the -expansion method was proposed to obtain the new exact solutions of the nonlinear evolution equations [24]. Subsequently the powerful -expansion method has been widely used by many differential such as in [25-30]. However, the previous works have mainly concentrated on obtaining new exact traveling wave solutions for the nonlinear evolution equations.

In this paper, by using the -expansion method, we construct non-traveling wave solutions with arbitrary functions in the (2+1)-dimensional breaking soliton system

(1)

(2)

where is an arbitrary constant, the system (1)-(2) was used to describes the (2+1)-dimensional interaction of Riemann wave propagated along the -axis with long wave propagated along the -axis and it seems to have been investigated extensively where overlapping solutions have been derived [31]. In the past, we have obtained the Annihilation solitons and chaotic solitons by the improved mapping approach [32]. Since the detailed physical background of the breaking soliton system has been given in [31], we neglect the corresponding description.

2. The (G'/G)-Expansion Method and Non-Traveling Wave Solutions to the (2+1)-Dimensional Breaking Soliton System

Before starting to apply the -expansion method, we will give a simple description of the method. For doing this, suppose that a (2+1)-dimensional nonlinear equation, say in three independent variables, y and t, is given by

(3)

The fundamental idea of the -expansion method is that the solutions of equation (3) can be expressed by a polynomial in as follows [24,29,30]:

(4)

where, is traveling wave transformation, and are constants to be determined later, G satisfies the second order LODE as follow:

(5)

In order to construct the non-traveling wave solutions with arbitrary function for the (2+1)-dimensional breaking soliton system (1)-(2), we suppose its solutions can be express as follow:

(6)

(7)

where, are the functions of to be determined later,

is the arbitrary function of x, y, t, and satisfies the second order LODE (5).

Applying the homogenous balance principle, we obtain. Thus (6)-(7) can be converted into

(8)

(9)

For simplifying the computation, we seek for the variable separation solutions of the breaking soliton system (2) by taking.

Substituting (8)-(9) into the system (1)-(2), collecting all terms with the same power of together. Then setting each coefficient of the polynomials to zero, we can derive a set of over-determined partial differential equation for and.

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

Solving the Equations (10)-(19) yields

(20)

Substituting (20) and the general solutions of Equation

(5) into (6)-(7), we can obtain the general non-traveling wave solutions for the (2+1)-dimensional breaking soliton system (1)-(2).

Case 1. When, by the general solutions of Equation (5) we can derive

(21)

Thus, the hyperbolic solutions of the system (1)-(2) are expressed as follows:

(22)

(23)

where, , , and existand, , and are arbitrary constants.

Case 2. When, by the general solutions of Equation (5) we can derive

(24)

So the trigonometric solutions of the system (1)-(2) are expressed as follows:

(25)

(26)

where, , , and existand, , and are arbitrary constants.

Case 3. When, by the general solutions of Equation (5) we can derive

(27)

In this case, the rational solutions of the system (1)-(2) are showed as:

(28)

(29)

where, , and exist, and, , and are arbitrary constants.

3. Soliton Structure Excitation of the System (1)-(2)

Due to the arbitrary functions and in the solutions (22)-(29), it is convenient to excite abundant soliton structures. We take the solution (23) as an example to study the soliton excitations for the (2+1)- dimensional breaking soliton system (1)-(2). For instance, if we choose and as

(30)

where are arbitrary constants, and all are non-zero.

Substituting (30) into (23) leads to a soliton structure for the system (1)-(2). Figures 1(a)-(d) are the evolution plots of the solution (23) with time under the parameters as

(31)

Figures 2(a)-(b) are the plots with, and we choose the following special values of the parameters. There are

(32)

(33)

Above we show the excitation process of a special dromion soliton structure of the solution (23) for the (2+1)- dimensional breaking soliton system (1)-(2). It is clear

that other selections of the arbitrary and in (23) may generate rich localized soliton structures. On the other hand, the solutions (22), (25)- (26), (28)-(29) may also be used to excite abundant soliton structures.

4. Summary and Discussion

In summary, via extending the -expansion method, more rich types explicit and exact non-traveling wave solutions of the (2+1)-dimensional breaking soliton system (1)-(2) are found out, and the traveling wave solutions are included by these non-traveling wave solutions. So the non-traveling wave solutions are more general. Furthermore, by choosing appropriately the arbitrary function included in its solutions, one can study various interesting localized soliton excitations. Since the wide applications of the soliton theory, to learn more about the localized excitations and their applications in reality is worthy of study further.

5. Acknowledgements

The authors would like to thank professor Zheng-yi Ma for his fruitful and helpful suggestions. The Scientific Research Foundation of Lishui University, China (Grant No. KZ201110).

REFERENCES