Degeneration of solitons for a ($$3+1$$)-dimensional generalized nonlinear evolution equation for shallow water waves

Long-Xing Li

doi:10.1007/s11071-022-07270-4

Abstract

A ($3+1$)-dimensional generalized shallow water waves equation is investigated with different methods. Based on symbolic computation and Hirota bilinear form, N-soliton solutions are constructed. In the process of degeneration of N-soliton solutions, T-breathers are derived by taking complexication method. Then rogue waves will emerge during the degeneration of breathers by taking the parameter limit method. Through full degeneration of N-soliton, M-lump solutions are derived based on long-wave limit approach. In addition, we also find out that the partial degeneration of N-soliton process can generate the hybrid solutions composed of soliton, breather and lump.

Highlights

A the (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves is investigated with different methods
Through full degeneration of N -soliton, M -lump solutions are derived based on long wave limit approach
N -soliton solutions are constructed based on symbolic computation and Hirota bilinear form

Summary

Introduction

We mainly investigate the (3+1)-dimensional generalized nonlinear evolution equation for the shallow water waves [35,36], the equation reads l1uxz + (l2ut + l3uxxx + l4uux)y + l5(ux∂x−1uy)x = 0. Based on the previous researches, we mainly investigate the deformation behaviors of N -soliton, most of solutions solutions obtained for Eq(1) are new and may be very meaningful to explain some special nonlinear science phenomenon.

From N -soliton solutions to T -order breather solutions

Evolution and degeneration from soliton to Lump solution

Full degeneration of N -solitons to M -lump solutions

The hybrid solutions

Conclusion