Some remarks on a linearized Schrödinger equation

Abstract

In the present paper we propose four systems of linear Partial Differential Equations that can be deduced from the nonlinear Schrödinger equation for the propagation of light in optical fibers in the frame of the recently-proposed Combined Regular-Logarithmic Perturbation method. The unknown function in the Schrödinger equation is the optical field envelope; it is a complex-valued function. Following the Combined Regular-Logarithmic Perturbation method, proposed by Secondini, Forestieri and Menyuk, we look for complex solutions of the Schrödinger equation in the form of a perturbed continuous wave that relates three unknown real-valued functions. Since the Schrödinger equation is complex, we split it into two real equations, both in the three real unknowns. We linearize these two equations and add a third linear equation that relates the same three unknown quantities. We propose four different choices for the third equation, therefore we obtain four di¤erent real systems of linear Partial Di¤erential Equations and we analyze the corresponding systems of Ordinary Differential Equations for the Fourier transforms of the unknowns. One of the four systems we obtain is equivalent to that studied by the quoted authors. We add to it other three choices that could be useful to model di¤erent situations. Again, we consider the real part of the Ordinary Differential Equations and we present solutions in recursive form. We also suggest solutions for the complex-valued Fourier transforms by using Bessel functions.