Abstract
Lie symmetry analysis is performed on a generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width equation. The symmetries and adjoint representations for this equation are given and an optimal system of one-dimensional subalgebras is derived. The similarity reductions and exact solutions with the aid ofG′/G-expansion method are obtained based on the optimal systems of one-dimensional subalgebras. Finally conservation laws are constructed by using the multiplier method.
Highlights
Nonlinear evolution equations (NLEEs) have been widely used to describe natural phenomena of science and engineering
Some of the well-known techniques used in the literature are the inverse scattering transform method [1], the homogeneous balance method [2], the Backlund transformation [3], the Weierstrass elliptic function expansion method [4], the Darboux transformation [5], the ansatz method [6, 7], Hirota’s bilinear method [8], the (G/G)-expansion method [9], the Jacobi elliptic function expansion method [10, 11], the variable separation approach [12], the sine-cosine method [13], the trifunction method [14, 15], the F-expansion method [16], the exp-function method [17], the multiple exp-function method [18], and the Lie symmetry method [19,20,21,22,23,24,25]
The purpose of this paper is to study one such NLEE, namely, the generalized two-dimensional nonlinear Kadomtsev-Petviashvili-modified equal width (KP-MEW) equation [26] that is given by (ut + αx + βuxxt)x + γuyy = 0
Summary
Nonlinear evolution equations (NLEEs) have been widely used to describe natural phenomena of science and engineering. In this paper we obtain symmetry reductions of (1) using Lie group analysis [19,20,21,22,23,24] and based on the optimal systems of one-dimensional subalgebras. In this subsection we use the optimal system of one-dimensional subalgebras calculated above to obtain symmetry reductions and exact solutions of the KP-MEW equation. Which gives rise to a group-invariant solution F = F(z) Using these invariants, (8) is transformed into the fourth-order nonlinear ODE:. After integrating and reverting back to the original variables, we obtain the following group-invariant solutions of the KP-MEW equation (1) for arbitrary values of n in the following form:. Which gives rise to a group-invariant solution F = F(r) and using these invariants, (17) is transformed to a second-order Cauchy-Euler ODE:.
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