Madelung Representation of Damped Parametric Quantum Oscillator and Exactly Solvable Schrödinger-Burgers Equations

Abstract

We construct a Madelung fluid model with time variable parameters as a dissipative quantum fluid and linearize it in terms of Schrödinger equation with time-dependent parameters. It allows us to find exact solutions of the nonlinear Madelung system in terms of solutions of the Schrödinger equation and the corresponding classical linear ordinary differential equation with variable frequency and damping. For the complex velocity field, the Madelung system takes the form of a nonlinear complex Schrödinger-Burgers equation, for which we obtain exact solutions using complex Cole-Hopf transformation. In particular, we give exact results for nonlinear Madelung systems related with Caldirola-Kanai-type dissipative harmonic oscillator. Collapse of the wave function in dissipative models and possible implications for the quantum cosmology are discussed. © 2010 American Institute of Physics.