Localized Nonlinear Waves on Spatio-Temporally Controllable Backgrounds for a (3+1)-Dimensional Kadomtsev-Petviashvili-Boussinesq Model in Water Waves

TL;DR

This paper develops a method to construct and analyze nonlinear water wave solutions on controllable spatio-temporal backgrounds using auto-Bäcklund transformations, revealing new ways to manipulate wave dynamics in inhomogeneous media.

Contribution

It introduces a novel approach to generate and control nonlinear wave solutions on variable backgrounds in a (3+1)-dimensional water wave model using simple mathematical tools.

Findings

Constructed kink-soliton and rogue wave solutions on controllable backgrounds.

Demonstrated modulation of nonlinear waves with periodic and localized backgrounds.

Provided graphical analysis of wave dynamics on inhomogeneous media.

Abstract

Physics of nonlinear waves on variable backgrounds and the relevant mathematical analysis continues to be the challenging aspect of the study. In this work, we consider a (3+1)-dimensional nonlinear model describing the dynamics of {water waves} and construct nonlinear wave solutions on spatio-temporally controllable backgrounds for the first time by using a simple mathematical tool auto-B\"acklund transformation. Mainly, we unravel physically interesting features to control and manipulate the dynamics of nonlinear waves through the background. Adapting an exponential function and general polynomial of degree two as initial seed solutions, we construct single kink-soliton and rogue wave, respectively. We choose arbitrary periodic, localized and combined wave backgrounds by incorporating Jacobi elliptic functions and investigate the modulation of these two nonlinear waves with a clear analysis and graphical demonstrations. The solutions derived in this work give us sufficient freedom to generate exotic nonlinear coherent structures on variable backgrounds and open up an interesting direction to explore the dynamics of various other nonlinear waves propagating through inhomogeneous media.