Madelung Representation and Exactly Solvable Schrodinger-Burgers Equations with Variable Parameters

Abstract

We construct a Madelung fluid model with specific time variable parameters as dissipative quantum fluid and linearize it in terms of Schrodinger equation with time dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrodinger equation and the corresponding classical linear ODE with variable frequency and damping. For the complex velocity field the Madelung system takes the form of a nonlinear complex Schrodinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we discuss and give exact results for nonlinear Madelung systems related with Caldirola-Kanai type dissipative harmonic oscillator.