Quadratic-Argument Approach to Nonlinear Schrodinger Equation and Coupled Ones

TL;DR

This paper introduces a quadratic-argument approach to solving coupled nonlinear Schrödinger equations, providing new exact solutions that encompass many known soliton solutions in nonlinear optics.

Contribution

The paper develops a novel quadratic-argument method for solving coupled nonlinear Schrödinger equations, expanding the set of exact solutions available for nonlinear optical wave interactions.

Findings

Derived new families of exact solutions including solitons

Unified known solutions as special cases of the new solutions

Enhanced understanding of wave interactions in nonlinear media

Abstract

The two-dimensional cubic nonlinear Schr\"{o}dinger equation is used to describe the propagation of an intense laser beam through a medium with Kerr nonlinearity. The coupled two-dimensional cubic nonlinear Schr\"{o}dinger equations are used to describe interaction of electromagnetic waves with different polarizations in nonlinear optics. In this paper, we solve the above equations by imposing a quadratic condition on the related argument functions and using their symmetry transformations. More complete families of exact solutions of such type are obtained. Many known interesting solutions, such soliton ones, turn out to be special cases of our solutions.