Resonant Soliton and Soliton‐Cnoidal Wave Solutions for a (3+1)‐Dimensional Korteweg‐de Vries‐Like Equation

Abstract

The residual symmetry of a (3+1)-dimensional Korteweg-de Vries (KdV)-like equation is constructed using the truncated Painlevé expansion. Such residual symmetry can be localized and the (3+1)-dimensional KdV-like equation is extended into an enlarged system by introducing some new variables. By using Lie’s first theorem, the finite transformation is obtained for this localized residual symmetry. Further, the linear superposition of multiple residual symmetries is localized and the n-th Bäcklund transformation in the form of the determinants is constructed for this equation. For illustration more detail, the first three multiple wave solutions-the collisions of resonant solitons are depicted. Finally, with the aid of the link between the consistent tanh expansion (CTE) method and the truncated Painlevé expansion, the explicit soliton-cnoidal wave interaction solution containing three kinds of Jacobian elliptic functions for this equation is derived.

Nonlocal symmetries and interaction solutions for the (n + 1)-dimensional generalized Korteweg–de Vries equation

“…When n = 2 and n = 3, the nonlocal symmetries and the finite symmetrical transformations for equation (1) have been constructed. Further, by the obtained symmetrical transformations, multiple soliton solutions for equation (1) with n = 2 and n = 3 have been obtained [37,38].…”

“…As an example, we give different types of soliton molecules as well as interaction solutions between multi-soliton and different periodic waves for equation (3). Only the given types of wave solutions of the (2+1) -or (3+1) -dimensional KdV equation can be seen from the existing literature [37][38][39], so the solutions in this paper are all new solutions.…”