Symmetry reduced and new exact non-traveling wave solutions of potential Kadomtsev–Petviashvili equation with p-power
Introduction
The potential Kadomtsev–Petviashvili(PKP) equation with p-power of nonlinearity (PKPp) studied in this paper is described aswhere . When and , 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. Letbe the symmetry of Eq. (1). To get some symmetries of Eq. (1), we take the function in the formwhere are functions to be determined.
- Case 1.
Here Eq. (1) becomes the following equationBased on Lie group theory [11], must satisfy the following equationand satisfies Eq. (3).
Reducing Eq. (3) by the symmetries (6)
- 3.1.1.
Let in (6), thenSolving the differential equation for getsSubstituting into Eq. (3), we get the following (1+1)D nonlinear PDE with constant coefficients:
- 3.1.2.
Taking in (6) yieldsSolving the differential equation for getsSubstituting (15) into Eq. (3), we have the function which must satisfy the following linear PDE:
- 3.1.3.
If we take
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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