Research paper
Symmetry reduction related with nonlocal symmetry for Gardner equation

https://doi.org/10.1016/j.cnsns.2016.06.017Get rights and content

Highlights

  • The nonlocal symmetry for the Gardner equation is derived by the truncated Painlevé analysis or the Möbious (conformal) invariant form.

  • To solve the initial value problem related with the nonlocal symmetry, the original Gardner equation is prolonged the enlarged systems.

  • Many explicit interaction solutions among different types of solutions are given by using the symmetry reduction method to the enlarged systems.

Abstract

Based on the truncated Painlevé method or the Möbious (conformal) invariant form, the nonlocal symmetry for the (1+1)–dimensional Gardner equation is derived. The nonlocal symmetry can be localized to the Lie point symmetry by introducing one new dependent variable. Thanks to the localization procedure, the finite symmetry transformations are obtained by solving the initial value problem of the prolonged systems. Furthermore, by using the symmetry reduction method to the enlarged systems, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, Painlevé II solutions are given. Especially, some special concrete soliton-cnoidal interaction solutions are analyzed both in analytical and graphical ways.

Introduction

In nonlinear science, the investigation of exact solutions for nonlinear evolution equations is one of the most important problems. Many effective methods have been proposed, such as the inverse scattering transformation [1], the Hirota’s bilinear method [2], symmetry reductions [3], the Darboux transformation [4], the Painlevé analysis method [5], the Bäklund transformation (BT) [6], the separated variable method [7], etc [8]. Among these traditional methods, it is difficult to obtain the interaction solutions among different nonlinear excitations [9]. However, the solitary waves must interact with other waves in the real physics world. How to find these interaction solutions is important topic in nonlinear science. Recently the localization procedure related with the nonlocal symmetry to find these types of interaction solutions has been proposed [10], [11], [12].

The aim of this paper is to explore the nonlocal symmetry of the Gardner equation and its applications. The finite symmetry transformations related with the nonlocal symmetry are obtained in the enlarged systems. The interaction solutions among solitons and other complicated waves including the Painlevé waves and periodic cnoidal waves of the Gardner equation are derived by the symmetry reduction method. Those interaction solutions are hard to obtain with other traditional methods.

The structure of this paper is as follows. In Section 2, the nonlocal symmetry for the Gardner equation is obtained with the truncated Painlevé method or the Möbious (conformal) invariant form. To obtain the finite symmetry transformations related by the nonlocal symmetry, the nonlocal symmetry for the original Gardner equation is localized by prolongation the Gardner equation. The finite symmetry transformations are thus obtained by solving the initial value problem of the Lie’s first principle. In Section 3, the symmetry reductions for the extended systems are considered according to the Lie point symmetry theory. The corresponding interaction solutions are given with the similar reductions. The last section is a short summary and discussion.

Section snippets

Nonlocal symmetry and its localization for Gardner equation

The (1+1)–dimensional Gardner equation reads ut+uxxx6β2u2ux+αuux=0,where α and β arbitrary constants [13]. (1) can be applied in dusty plasma, ocean and atmosphere, fluid mechanics and solid state physics [14], [15], [16]. Various methods for studying integrability properties and exact solutions of the Gardner equation have been reported. [17], [18], [19], [20], [21], [22], [23]. In this section, we shall study the nonlocal symmetry and its localization process for the Gardner Eq. (1).

Similarity reductions related with nonlocal symmetry

The nonlocal symmetry can not be used to construct explicit solutions by the symmetry reduction directly. By introducing the potential field (10), the nonlocal symmetry becomes the usual Lie point symmetry in the prolonged systems. We can thus use the symmetry reduction related with the nonlocal symmetry to study the the prolonged systems. These symmetry reduction solutions can not be obtained within the framework of the direct Lie’s symmetry method. The Lie point symmetries σk(k=u,ϕ,g) for the

Conclusions

The nonlocal symmetry of the Gardner equation is given by the truncated Painlevé analysis or the Möbious invariant form. Then, the nonlocal symmetry is localized to the local Lie point symmetries for the prolonged system. With the help of the localization procedure, the nonlocal symmetry is used to find possible symmetry reductions. The rational solution hierarchy, soliton + Painlevé II wave solutions, soliton + cnoidal wave solutions and other combinational solutions of elliptic functions are

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 11305106.

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