Soliton solutions and traveling wave solutions for the two-dimensional generalized nonlinear Schr\"odinger equations

TL;DR

This paper investigates the integrability and explicit solutions of two-dimensional generalized nonlinear Schrödinger equations, deriving soliton and traveling wave solutions using Hirota's method and the extended tanh method, with applications to physical phenomena.

Contribution

It introduces bilinear forms and explicit soliton and traveling wave solutions for the 2D GNLS equations, demonstrating their integrability and physical relevance.

Findings

Derived one- and two-soliton solutions using Hirota's method

Constructed new exact traveling wave solutions with the extended tanh method

Visualized solution dynamics through 3D plots

Abstract

In this paper, we present the two-dimensional generalized nonlinear Schr\"odinger equations with the Lax pair. These equations are related to many physical phenomena in the Bose-Einstein condensates, surface waves in deep water and nonlinear optics. The existence of the Lax pair defines integrability for the partial differential equation, so the two-dimensional generalized nonlinear Schr\"odinger equations are integrable. We obtain bilinear forms of the two-dimensional GNLS equations. One- and two-soliton solutions are derived via the Hirota bilinear method, a procedure quite useful in the solution of nonlinear partial differential equations. We apply the extended tanh method in order to construct new exact traveling wave solutions. Through 3D plots, we show the dynamical behavior of the obtained solutions.