Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation

TL;DR

This paper introduces new solutions and rogue wave phenomena in the (2+1)-dimensional Korteweg-de Vries equation using bilinear transformations, Lax pairs, and arbitrary functions, revealing complex dynamic behaviors.

Contribution

It presents novel bilinear Backlund transformation, Lax pair, and deformed rogue wave solutions for the (2+1)-dimensional KdV equation, expanding understanding of 2D rogue waves.

Findings

Deformed 2D rogue waves resemble Peregrine soliton on soliton plane

New bilinear Backlund transformation and Lax pair derived

Dynamic behaviors of solutions visualized in 3D plots

Abstract

Within the (2 + 1)-dimensional Korteweg-de Vries equation framework, new bilinear Backlund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function, a family of deformed soliton and deformed breather solutions are presented with the improved Hirotas bilinear method. Choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of obtained solitons. Additionally, two dimensional [2D] rogue waves (localized in both space and time) on the soliton plane are presented, we refer to it as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function. This new idea is also applicable to other nonlinear systems.