Traveling wave solutions for the generalized (2+1)-dimensional Kundu-Mukherjee-Naskar equation

TL;DR

This paper analyzes the bifurcation and phase portraits of traveling wave solutions for the generalized (2+1)-dimensional Kundu-Mukherjee-Naskar equation, deriving explicit solutions through elliptic integrals.

Contribution

It introduces a detailed bifurcation analysis and explicit traveling wave solutions for the generalized Kundu-Mukherjee-Naskar equation using dynamical systems and elliptic integrals.

Findings

Global phase portraits for different parameter sets

Explicit type I and type II traveling wave solutions

Complete classification of bounded and unbounded orbits

Abstract

In this paper, we consider two types of traveling wave systems of the generalized Kundu-Mukherjee-Naskar equation. Firstly, due to the integrity, we obtain their energy functions. Then, the dynamical system method is applied to study bifurcation behaviours of the two types of traveling wave systems to obtain corresponding global phase portraits in different parameter bifurcation sets. According to them, every bounded and unbounded orbits can be identified clearly and investigated carefully which correspond to various traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation exactly. Finally, by integrating along these orbits and calculating some complicated elliptic integral, we obtain all type I and type II traveling wave solutions of the generalized Kundu-Mukherjee-Naskar equation without loss.