Abstract
In this work, we investigate a (3+1)-dimensional generalised Kadomtsev–Petviashvili equation, recently introduced in the literature. We determine its group invariant solutions by employing Lie symmetry methods and obtain elliptic, rational and logarithmic solutions. The solutions derived in this paper are the most general since they contain elliptic functions. Finally, we derive the conserved quantities of this equation by employing two approaches—the general multiplier approach and Ibragimov’s theorem. The importance of conservation laws is explained in the introduction. It should be pointed out that the investigation of higher dimensional nonlinear partial differential equations is vital to our perception of the real world since they are more realistic models of natural and man-made phenomena.
Highlights
The study of nonlinear partial differential equations (NLPDEs) and their solutions has become a subject of much interest in the past few decades
Along with the progress in modelling nonlinear phenomena came a myriad of methods designed to derive the exact solutions of these models
The first solution contains an elliptic integral of the second kind, an amplitude function, a delta amplitude function and a Jacobi cosine function
Summary
The study of nonlinear partial differential equations (NLPDEs) and their solutions has become a subject of much interest in the past few decades. In their work [29], the authors showed that addition of the new term affects the dispersion relations significantly They applied Hirota’s direct method to determine the multiple soliton solutions of (3). We seek to derive the exact solutions of the (3+1)-D gKPe (3) by making use of its Lie point symmetries and direct integration. Lie perceived that the seemingly different methods for finding exact solutions of differential equations were, in reality, all special cases of a broad integration approach; the theory of transformation groups. This theory is an analog of Galois theory and has an enormous impact on mathematics and mathematical physics today. The ODE (10) describes stationary waves and by imposing certain constraints such as having the fluid undisturbed at infinity, Korteweg and de Vries obtained negative and positive solitary waves as well as cnoidal wave solutions [49,50]
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