Lie symmetry analysis and exact explicit solutions of three-dimensional Kudryashov–Sinelshchikov equation

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Highlights

  • All of the geometric vector fields of the equation are obtained.

  • The symmetry reductions are presented.

  • We obtain some new exact explicit solutions.

Abstract

In this paper, the three-dimensional Kudryashov–Sinelshchikov (KS) equation, which describes the propagation of nonlinear waves in a bubbly liquid. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained, the symmetry reductions and some new exact explicit solutions are also presented.

Introduction

In 2010, Kudryashov and Sinelshchikov presented the following nonlinear partial differential equation [1]:ut+αuux+uxxx-(uuxx)x-βuxuxx=0,where u is a density and which model heat transfer and viscosity, α,β are real parameters. Eq. (1) is called Kudryashov–Sinelshchikov equation and is for describing the pressure waves in a mixture liquid and gas bubbles taking into consideration the viscosity of liquid and the heat transfer, it is generalization of the KdV and the BKdV equation and similar but not identical to the Camassa–Holm equation. Eq. (1) was studied by many researchers in various methods[2], [3], [4], [5], [6], [7], [8], [9]. Further, in the more realistic situations, the following three-dimensional Kudryashov–Sinelshchikov equation [10](ut+uux+uxxx-χuxx)x+12(uyy+uzz)=0has been considered to describe the physical characteristics of nonlinear waves in a bubbly liquid, where u represents the density of the bubbly liquid, the scalar quantity χ depends on the kinematic viscosity of the bubbly liquid, x,y and z are the scaled space coordinates, t is the scaled time coordinate, and the subscripts denote the partial derivatives. In the case without the last term, Eq. (2) reduces to the two-dimensional KdV–Burgers equation, while simultaneously χ=0 (i.e. the liquid is ideal), to the Kudryashov–Sinelshchikov equation. Painlevé analysis has been carried out on Eq. (2), which shows that Eq. (2) is not Painlevé integrable [10]. One-and two-solitary wave solutions of Eq. (2) have been given and stability of the solitary waves without the dissipative losses has been studied [10]. To Eq. (2), the periodic, solitary wave solutions for χ=0, and kink wave solutions for χ0 are obtained by Bifurcations and phase portraits in [11].

The main purpose of this paper is to investigate the vector fields, the symmetry reductions and exact solutions to Eq. (2) by means of the combination of Lie symmetry analysis [12], [13], [14], [15], [16] and the Riccati equation method [17], [18]. The rest of this paper is organized as follows: in Section 2, the Lie symmetry analysis is performed on Eq. (2), the complete geometric vector fields of the equation are obtained. In Section 3, different types of symmetry reductions of Eq. (2) are obtained. In Section 4, some new exact explicit solutions are presented. The last section is a short summary and discussion.

Section snippets

Lie symmetries for Eq. (2)

First of all, let us consider a one-parameter Lie group of infinitesimal transformation:xx+ξ(x,y,z,t,u),yy+η(x,y,z,t,u),zz+μ(x,y,z,t,u),tt+τ(x,y,z,t,u),uu+ϕ(x,y,z,t,u),with a small parameter 1. The vector field associated with the above group of transformations can be written asV=ξ(x,y,z,t,u)x+η(x,y,z,t,u)y+μ(x,y,z,t,u)z+τ(x,y,z,t,u)t+ϕ(x,y,z,t,u)u.The symmetry group of Eq. (2) will be generated by the vector field of the form (3). Applying the fourth prolongation pr(4)V

Symmetry reductions

In this section we will obtain symmetry reductions of Eq. (2) by means of the symmetry analysis.

Exact explicit solutions

Obviously, it is easier for us to seek the explicit solutions to the reduction equations than to solve the (3+1)-dimensional KS equation. For example, we will consider the traveling wave solutions of Eq. (21), Eq. (30) with variable coefficients and Eq. (46) by using the Riccati equation method [17], [18] in this section.

Using a traveling wave variable of Eq. (21) asU(X,y,z)=v(ω),ω=kX+ly+mz,where X=x-t and k,l,m are constants. Substituting (62) into (21), then Eq. (21) is reduced to the

Conclusions

In this paper, the invariance propertied of the (3+1)-dimensional Kudryashov–Sinelshchikov equation are presented by using the Lie symmetry analysis. All of the geometric vector fields and the symmetry reductions of the (3+1)-dimensional KS equation are obtained. Furthermore, some exact explicit solutions are obtained by considering one of the reductions. In fact, more exact explicit solutions to the (3+1)-dimensional KS equations can be derived through other reduction equations. Because of the

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Nos. 11461022, 11361023) and the Major Natural Science Foundation of Yunnan Province, China (No. 2014FA037).

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