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

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.

💡 Research Summary

The paper develops a comprehensive framework for describing quantum fluids with time‑dependent dissipative and oscillatory parameters by extending the Madelung representation. Starting from the standard Madelung transformation, the authors rewrite the wavefunction (\psi(\mathbf{r},t)=\sqrt{\rho(\mathbf{r},t)},e^{iS(\mathbf{r},t)/\hbar}) in terms of a real density (\rho) and a real phase (S). Unlike the conventional treatment, they allow the mass (m(t)), the damping coefficient (\gamma(t)), and the frequency (\omega(t)) to vary arbitrarily with time. Substituting this ansatz into a Schrödinger equation with these time‑dependent parameters yields a pair of coupled nonlinear partial differential equations: a continuity equation with a loss term (-\gamma(t)\rho) and a quantum‑Bernoulli equation containing the quantum potential.

The key insight is that these two equations can be combined into a single complex equation for a complex velocity field
\