Investigation of a new similarity solution for the (2+1)-dimensional Schwarzian Korteweg–de Vries equation

Abstract

We study the (2+1)-dimensional Schwarzian Korteweg–de Vries equation (SKdV). The explored solutions describe new Lump soliton colliding with visible soliton, an interaction between multi-soliton waves with one soliton, multi peaks of waves moving in a curved path, two hyperbolic waves moving together without interaction and some of periodic waves. We examine the commutative product between multi unknown Lie infinitesimals for the (2+1)-dimensional (SKdV) equation, and this study result in some new Lie vectors. The commutative product generates a system of nonlinear ODEs which had been solved manually. Through two stages of Lie symmetry reduction, SKdV equation is reduced to non-solvable nonlinear ODEs using various combinations of optimal Lie vectors. Using the Riccati–Bernoulli sub-ODE and Integration methods, we investigate new analytical solutions for these ODEs. Back substituting for the original variables generates new solutions for SKdV. Some selected solutions are illustrated through three-dimensional plots.