Higher-order rational solitons and rogue-like wave solutions of the (2+1)-dimensional nonlinear fluid mechanics equations

TL;DR

This paper develops generalized Darboux transformations to derive higher-order rational solitons and rogue wave solutions for the (2+1)-dimensional KP equation, revealing complex wave structures and parameter-controlled dynamics.

Contribution

It introduces a novel method using generalized perturbation Darboux transformations to obtain explicit rogue wave and soliton solutions for the KP equation.

Findings

Discovery of diverse rogue wave structures like triangles and pentagons.

Parameter (a) controls transition between rogue waves and rational solitons.

Wave dynamics depend on parameters and time, showing complex behaviors.

Abstract

The novel generalized perturbation (n, M)-fold Darboux transformations (DTs) are reported for the (2+1)-dimensional Kadomtsev-Petviashvili (KP) equation and its extension by using the Taylor expansion of the Darboux matrix. The generalized perturbation (1, N-1)-fold DTs are used to find their higher-order rational solitons and rogue wave solutions in terms of determinants. The dynamics behaviors of these rogue waves are discussed in detail for different parameters and time, which display the interesting RW and soliton structures including the triangle, pentagon, heptagon profiles, etc. Moreover, we find that a new phenomenon that the parameter (a) can control the wave structures of the KP equation from the higher-order rogue waves (a>0 or a<0) into higher-order rational solitons (a = 0) in (x, t)-space with y=const. These results may predict the corresponding dynamical phenomena in the models of fluid mechanics and other physically relevant systems.