Self-similarity analysis of the non-linear Schr\"odinger equation in the Madelung form

TL;DR

This paper analyzes the non-linear Schrödinger equation in Madelung form, revealing how increased nonlinearity causes greater deviation from linear solutions, using a hydrodynamic analogy.

Contribution

It reformulates a specific non-linear Schrödinger equation into a hydrodynamic form and studies the effects of nonlinearity on solution behavior.

Findings

Higher nonlinear coefficients lead to stronger deviations from linear solutions.

The Madelung transformation provides a hydrodynamic perspective on the non-linear Schrödinger equation.

Nonlinear effects become more pronounced with increasing nonlinearity.

Abstract

In the present study a particular case of Gross-Pitaevskii or non-linear Schr\"odinger equation is rewritten to a form similar to a hydrodynamic Euler equation using the Madelung transformation. The obtained system of differential equations is highly nonlinear. Regarding solutions, a larger coefficient of the nonlinear term yields stronger deviation of the solution from the linear case.