Applied Mathematics
Vol.05 No.20(2014), Article ID:51586,6 pages

Symmetry Reduction and Explicit Solutions of the (2 + 1)-Dimensional DLW Equation

Zhengyi Ma1,2, Jinxi Fei1, Yuanming Chen1

1Department of Mathematics, Zhejiang Lishui University, Lishui, China

2Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai, China


Copyright © 2014 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 27 August 2014; revised 25 September 2014; accepted 18 October 2014


Utilizing the Clarkson-Kruskal direct method, the symmetry of the (2 + 1)-dimensional dispersive long wave equation is derived. From which, through solving the characteristic equations, four types of the explicit reduction solutions that related the hyperbolic tangent function are obtained. Finally, several soliton excitations are depicted from one of the solutions.


dispersive long Wave Equation, symmetry Reduction, Explicit solution,soliton Excitation

1. Introduction

Soliton theory, one of the typical topics in nonlinear science, has been widely applied in optics of nonlinear media, photonics, plasmas, mean-field theory of Bose-Einstein condensates, condensed matter physics, and many other fields. For describing these nonlinear physical phenomena, the study of symmetry is a very important approach, especially in the integrable nonlinear partial differential equations (NPDEs) for the sake of the existence of symmetries in infinity. Traditionally, there are three powerful methods to find the symmetry structure of the NPDEs, that is, the Lie group method of infinitesimal transformations, the nonclassical Lie group method and the Clarkson and Kruskal (CK) direct method [1] - [5] . Among them, the classical Lie symmetries of the partial differential equations (PDEs) can be obtained through the Lie group method of infinitesimal transformations. Using the basic prolongation method and the infinitesimal criterion of invariance, one can find some particular Lie point symmetries group of the NPDEs.

In Section 2 of this paper, a (2 + 1)-dimensional dispersive long wave (DLW) equation is taken to illustrate the symmetry reduction related the CK direct method. Section 3 is a direct result which the explicit reduction solutions are solved and the soliton excitations are depicted. Section 4 is the conclusion.

2. Symmetry Structure through the Direct Approach

In the following of this paper, we fucus on the (2 + 1)-dimensional dispersive long wave (DLW) equation


The system was first derived by Boiti et al. as a compatibility for a weak Lax pair [6] . In Ref. [7] , Paquin and Winternitz showed that the symmetry algebra of Equation (1) is infinite-dimensional and a Kac-Moody-Virasoro structure. The more general symmetry algebra, symmetry algebra, was given in Ref. [8] . In Ref. [9] , Lou gave out nine types of two dimensional similarity reductions and thirteen types of ordinary differential equation reductions. In Ref. [10] , Lou showed that Equation (1) has no Painléve property, though the system is Lax or 1ST integrable. Abundant propagating localized excitations were also derived by Lou [11] [12] with the help of Painlvé-Bäcklund transformation and a multilinear variable separation approach. With the aid of a projective Riccati equation approach and by introducing appropriate lower-dimensional localized patterns, abundant coherent soliton excitations, that is, solitons, chaos and fractals were derived by ours [13] - [15] .

According to the Clarkson and Kruskal (CK) direct method [4] [5] , we first seek the similarity reduction of Equation (1) in the form of


where, and, , are all

differentiable functions to be determined, , satisfy the following DLW equation

as Equation (1)


The result of the symbolic computation, one can deduce


where and are arbitrary functions of their own variables.

Utilizing the variable separation solution of Equation (3) which derived in Refs. [13] [14]


the similarity solution of Equation (1) can be written


where and are two arbitrary variable separation functions of and, respectively.

Second, under the transformation


with the infinitesimal parameter, we have



then Equation (2) can reduce to




The equivalent vector expression of the above symmetry is


When taking


from Equation (13), the following six operators are obtained


Hence, we obtain the commutator table listed in Table 1 with the -th entry indicating

according to the commutator operators.

3. Reduction Solutions

Solving the following characteristic equations


with, , , and

, we can obtain four types of similarity reductions of Equation (1).

1) Taking we get the similarity reduction of Equation (1) through solving Equations (16)



2) Taking we get the similarity reduction of Equation (1) through solving Equations (16)

Table 1. Lie Bracket.



3) Taking we get the similarity reduction of Equation (1) through solving

Equations (16)


4) Taking we get the similarity reduction of Equation (1) through solving Equations (16)


As we all know, to derive soliton structures of a explicit solution is a meaningful task fo a nonlinear physical

equation. Now, when taking (is a solution in Equation (17)), a fundmental soliton, that is a

dromion-like structure, is found if the constants, , the function

and the fixed time (Figure 1(a)). A further considering is, when the cons- tants and the function,. For this time, a typical line soliton is derived for the fixed time (Figure 1(b)).

The solitoff is another special type of coherernt structure for a nonlinear equation, where the wave fields

decays exponentially in all directions except for a preferred direction [16] . For the field, the solitoff

structure can also be constructed. Figure 2(a) shows a two-solitoff solution when taking the constants

, , the function and

. After adjusting the constants and the

(a) (b)

Figure 1. (a) A dromion structure of the solution (17) when taking the constants and the function ; (b) The line soliton structure of the solution (17) when taking the constants and the function.

(a) (b)

Figure 2. (a) A two-solitoff soliton of the solution (17) when taking the constants, and the function; (b) The periodic dromion solitons of the solution (17) when taking the constants and the function.

function, the field, the periodic dromion

solitons is depicted (Figure 2(b)).

4. Summary and Conclusion

In summary, we have obtained the symmetry reduction with the aid of the CK direct method and some explicit solutions through solving the characteristic equations of the (2 + 1)-dimensional DLW equation. These obtained solutions contain several free functions of variables and, which provide us with more chose of these functions to generate the abundant soliton structures. Here, we chose several types of elementary functions to exhibit these soliton propagations related to the obtained solutions. These solutions may provide more infor- mation to further study the nonlinear physical system.


The authors are grateful to Profs Lou S. Y. and Chen Y. and Drs Xin X. P. and Hu X. X. for their helpful suggestions and fruitful discussion.


Supported by the Natural Science Foundation of Zhejiang Province, China under Grant Nos. LY14A010005 and LQ13A010013.

Cite this paper

Zhengyi Ma,Jinxi Fei,Yuanming Chen, (2014) Symmetry Reduction and Explicit Solutions of the (2 + 1)-Dimensional DLW Equation. Applied Mathematics,05,3264-3269. doi: 10.4236/am.2014.520304


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