Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

Abstract

In this article, (3+1)-dimensional generalized Shallow Water-like (gSWl) equation is discussed. The infinitesimal generators of the equation are derived by using the Lie symmetry analysis method. The optimal system is obtained based on the adjoint table of the generators of the equation. Exact solutions of the equation are constructed by applying symmetry reduction, Exp−ϕ(ξ) expansion method, Exp-function expansion method, Riccati equation method, and G′/G expansion method. For analyzing the dynamical behavior of the solutions, we derive the physical structures of dark soliton, kink wave, and periodic solutions via numerical simulations.

Non-local residual symmetry and soliton-cnoidal periodic wave interaction solutions of the KdV6 equation

“…As is known, symmetry reduction [25,26] is one of the most powerful methods to study exact explicit solutions for non-linear equations. Based on the Lie point symmetry (12), one may continue to explore more abundant symmetry reduction solutions for the KdV6 equation.…”