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

TL;DR

This paper develops a Madelung fluid model with variable parameters, linearizes it via a time-dependent Schrödinger equation, and finds exact solutions for complex velocity fields and dissipative harmonic oscillators.

Contribution

It introduces a novel Madelung fluid model with variable parameters and derives exact solutions for the nonlinear Schrödinger-Burgers system using complex Cole-Hopf transformation.

Findings

Exact solutions for Madelung systems with variable parameters.

Analytical solutions for dissipative harmonic oscillators.

Extension of Schrödinger equation methods to nonlinear fluid models.

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.