Maxwell's equations and the vector nonlinear Schrödinger equation

Abstract

We examine similarities and fundamental differences between Maxwell's equations and the vector nonlinear Schr\"odinger equation (which is an approximation of the former) in describing a light evolution in a uniform nonlinear anisotropic medium. It is shown that in some cases, the solitary wave solutions to the nonlinear Schr\"odinger equation cannot be recovered from Maxwell"s equations while in others the solitary wave solutions to Maxwell's equations are lost from the nonlinear Schr\"odinger equation through approximation (even in the limit under which the approximation is derived or valid), although there are cases where the solutions to the two sets of equations demonstrate only quantitative differences. The existence of novel classes of the hybrid vector solitary waves composed of three field components is also demonstrated and the bifurcation characteristics of the solitary wave states are analyzed.