Elsevier

Applied Mathematics and Computation

Volume 268, 1 October 2015, Pages 432-438
Applied Mathematics and Computation

Symmetry reduction and explicit solutions of the (2+1)-dimensional Boiti–Leon–Pempinelli system

https://doi.org/10.1016/j.amc.2015.06.086Get rights and content

Abstract

Through the standard truncated Painlevé expansion of the Boiti–Leon–Pempinelli equation, the residual symmetry is localized in the properly prolonged system with the Lie point symmetry vector. Some different transformation invariances are derived through the obtained symmetries. At the same time, the symmetry of the equation is also derived utilizing the Clarkson–Kruskal direct method. From which, through solving the characteristic equations, several types of the explicit reduction solutions that related the hyperbolic tangent function are obtained. Finally, some kink-type soliton excitations are depicted from one of them.

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, one of many effective ways is to derive exact solutions including the soliton solutions and different wave solutions for a integrable system. The study of symmetry has been a very important approach to obtain the explicit solutions since the 1970s, 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], [2], [3], [4], [5].

Recently, an attractive work developed by Lou et al. [6], [7] which is called the residual symmetry, is proposed that the symmetry related to the Painlevé truncated expansion is just the residue with respect to the singular manifold. Considering the localizations of the residual symmetries and to find related finite transformations, many types of nonlocal symmetries can be localized to Lie point symmetries by introducing suitable prolonged systems [8], [9], [10], [11], [12]. At the same time, through the repeated symmetry reduction approach, the infinite many symmetries and exact solutions of Burgers equation are derived by Lou and Lian [13].

In Section 2 of this paper, a (2+1)-dimensional Boiti–Leon–Pempinelli (BLP) equation is taken to illustrate the residual related symmetry reduction approach. Section 3 is the symmetry reduction related the CK direct method for the BLP equation. The direct result is that, the explicit reduction solutions are solved and the soliton excitations are depicted. Section 4 is the conclusion.

Section snippets

The residual symmetry of the (2+1)-dimensional BLP equation

In the following of this paper, we focus on the (2+1)-dimensional Boiti–Leon–Pempinelli (BLP) equation uyt2uxuy2uuxy+uxxy2vxxx=0,vtvxx2uvx=0.This system has extensive physical background and its integrability and Hamiltonian were established by using the sine-Gordon and the sinh-Gordon equations [14], [15]. The special Painlevé Bäcklund transformation and multilinear variable separation approach were applied to get some special localized excitations with fractal behaviors [16]. Also, the

Symmetry reduction solutions of the (2+1)-dimensional BLP equation

Solving the following characteristic equations dxX=dyY=dtT=duΦ1=dvΦ2,with X=xf1,t(t)+f2(t),Y=f4(y),T=2f1(t),Φ1=f1,t(t)U+12(f1,tt(t)x+f2,t(t)) and Φ2=f4,y(y)V+f3(y), we can obtain the following types of similarity reductions of Eq. (1).

  • 1.

    Taking f1(t)=c1,f2(t)=c2, we get the similarity reduction of Eq. (1) through solving Eqs. (40) u1=B2A+Atanh(A(xc2Δ1)+B(t2c1Δ1)),v1=(Ac2+2Bc1)(tanh(A(xc2Δ1)+B(t2c1Δ1)))0yf3(s)ds+kf4(y),where Δ1=0ydsf4(s). Due to arbitrariness of functions f3(y) and f4(y)

Conclusion

In summary, we have obtained the residual related symmetry reduction of the (2+1)-dimensional Boiti– Leon–Pempinelli (BLP) equation and the symmetry reduction with the aid of the CK direct method. As a result, some explicit solutions through solving the characteristic equations of the (2+1)-dimensional BLP equation have been derived. These obtained solutions contain several free functions of variables x, y and t, which provide us with more choices of these functions to generate the abundant

Acknowledgments

The authors are grateful to Profs. S.Y. Lou and Y. Chen and Drs. X.P. Xin and X.X. Hu for their helpful suggestions and fruitful discussion. The work was supported by the National Natural Science Foundation of China under grant no. 11447017 and the Natural Science Foundation of Zhejiang Province of China under grant nos. LY14A010005 and LQ13A010013.

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