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