Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev–Petviashvili equation with p-power

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Abstract

With the aid of Maple symbolic computation and Lie group method, PKPp equation is reduced to some (1 + 1)-dimensional partial differential equations, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Using the separation of variables, homoclinic test technique and auxiliary equation methods, we obtain new abundant exact non-traveling solution with arbitrary functions for the PKPp.

Introduction

The potential Kadomtsev–Petviashvili(PKP) equation with p-power of nonlinearity (PKPp) studied in this paper is described asuxt+αuxpuxx+βuxxxx-γuyy=0,where u:Rx×Ry×Rt+R,α,β,γR,pN,α·γ0,γ=±1. When α=6,β=1,p=1 and γ=-1, Eq. (1) becomes the potential Kadomtsev–Petviashvili equation [1]. And its solutions have been studied extensively. By using various methods and techniques, exact traveling wave solutions, linearly solitary wave solutions, soliton-like solutions and some numerical solutions have been obtained for PKP equation [2], [3], [4], [5]. Recently, Dai etc. proposed a homoclinic test method for seeking solution of PKP. The exact periodic kink-wave solution, periodic soliton solution and doubly periodic solution of PKP equation are constructed [6], [7].

In this paper, by means of Maple symbolic computation, we will use the Lie group method, homoclinic test technique method etc. to reduce and solve PKPp(1) [7], [8], [9], [10], [11], [12]. First, we will seek symmetry of Eq. (1) in two cases. Then we use the symmetry to reduce Eq. (1) to some (1 + 1)-dimensional PDE, in which there are linear PDE with constant coefficients, nonlinear PDE with constant coefficients, and nonlinear PDE with variable coefficients. Finally, solving the reduced PDE with constant coefficients by Homoclinic test technique and auxiliary equation methods etc. implies abundant exact non-traveling wave periodic solutions for the PKPp equation.

Section snippets

Symmetry of Eq. (1)

In this section, we seek for symmetries of Eq. (1) in two cases. Letσ=σx,y,t,u,ux,uy,utbe the symmetry of Eq. (1). To get some symmetries of Eq. (1), we take the function σ in the formσ=a1(x,y,t)ut+a2(x,y,t)ux+a3(x,y,t)uy+a4(x,y,t)u+a5(x,y,t),where ai(x,y,t)(i=15) are functions to be determined.

  • Case 1.

    p=1

    Here Eq. (1) becomes the following equationuxt+αuxuxx+βuxxxx-γuyy=0.Based on Lie group theory [11], σ must satisfy the following equationσxt+αuxxσx+αuxσxx+βσxxxx-γσyy=0and u(x,y,t) satisfies Eq. (3).

Reducing Eq. (3) by the symmetries (6)

  • 3.1.1.

    Let c1=c3=c4=0,c2=1 in (6), thenσ=ut+λ(t)y+μ(t).Solving the differential equation for σ=0 getsu=-(λ(t)y+μ(t))dt+ϕ(x,y).Substituting into Eq. (3), we get the following (1+1)D nonlinear PDE with constant coefficients:αϕxϕxx+βϕxxxx-γϕyy=0.

  • 3.1.2.

    Taking c1=c2=c4=0,c3=1 in (6) yieldsσ=ux+λ(t)y+μ(t).Solving the differential equation for σ=0 getsu=-(yλ(t)+μ(t))x+ϕ(y,t).Substituting (15) into Eq. (3), we have the function ϕ(y,t) which must satisfy the following linear PDE:λ(t)y+μ(t)+γϕyy=0.

  • 3.1.3.

    If we take c4=1,c

Solutions of some new reduced PDE with constant coefficients

In this paper, we use some appropriate methods to solve reduced Eqs. (16), (19) with constant coefficients, then obtain some new explicit solutions of Eq. (1).

Conclusions

In this paper, the new idea, a combination of Lie group method and homoclinic test technique etc. is applied and thus the symmetries (6), (10) are obtained. The (2 + 1)-PKPp Eq. (1) is reduced to (1 + 1)-dimensional linear PDE (16), nonlinear PDE of constant coefficients (13), (19), (28), (31) and nonlinear PDE of variable coefficients (22), (25). Further Homoclinic test technique, Auxiliary Equation method, extended homogeneous balance method are used and some new exact non-traveling wave

Acknowledgements

We are very grateful to editors, reviewing scholars for their careful work on the manuscript and their constructive comments.

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