Self-similarity analysis of the non-linear Schrödinger equation in the Madelung form

Abstract

In the present study a particular case of Gross-Pitaevskii or non-linear Schrödinger 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.

11 pages, 2 figures, Accepted in Advances in Mathematical Physics