Lie transformations, similarity reduction, and solutions for the nonlinear Madelung fluid equations with external potential

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Abstract
The application of Lie-group methods to a system of coupled nonlinear partial differential equations representing what is usually called a Madelung fluid is shown. The generating operators of the transformation group that depends on five arbitrary group constants will be constructed, and all subclasses of systems of ordinary differential equations derived by similarity reduction will be presented in tabular form. Two subclasses of physical interest are investigated in detail and the similarity solutions are compared with solutions found earlier by the application of inverse scattering transform techniques to the cubic nonlinear Schrödinger equation. Similarity solutions for the Madelung equations with linear external potential Γ(x)=−f0x are presented.