Dual-mode nonlinear Schrödinger equation (DMNLSE): Lie group analysis, group invariant solutions, and conservation laws

Abstract

This paper proposed a novel dual-mode nonlinear Schrödinger equation (DMNLSE) with cubic law. The DMNLSE describes the propagations of two moving waves simultaneously. This nonlinear Schrödinger equation (NLSE) model appears in different areas of physics, including nonlinear optics, plasma physics, superconductivity, and quantum mechanics. In the context of photonics, NLSE models the propagation of soliton pulses through inter-continental distances. In this study, the Lie symmetry procedure is successfully utilized on the DMNLSE to discover the infinitesimal generators using the invariance condition. Then, we transform the DMNLSE into an ordinary differential equation (ODE) employing accepted generators and similarity reduction concepts. Later, we consider conservation laws for the DMNLSE. Thanks to the multiplier technique, the conserved quantities' densities and fluxes are discovered. With the aid of the Mathematica package program, 3D, contour, and density graphs were drawn for the special values of the parameters in the solutions, and the physical structures of the solutions obtained in this way were also observed.