Modulation of localized solutions for the Schr\"odinger equation with logarithm nonlinearity

TL;DR

This paper explores localized solutions of the Schrödinger equation with logarithmic nonlinearity, transforming inhomogeneous equations into autonomous forms and analyzing the stability of solutions.

Contribution

It introduces a method to find analytical localized solutions for inhomogeneous Schrödinger equations with logarithmic nonlinearity using similarity transformations.

Findings

Analytical localized solutions derived for inhomogeneous equations

Stability of solutions assessed numerically

Method applicable to similar nonlinear equations

Abstract

We investigate the presence of localized analytical solutions of the Schr\"odinger equation with logarithm nonlinearity. After including inhomogeneities in the linear and nonlinear coefficients, we use similarity transformation to convert the nonautonomous nonlinear equation into an autonomous one, which we solve analytically. In particular, we study stability of the analytical solutions numerically.