Exactly solvable Madelung fluid and complex Burgers equations: a quantum Sturm–Liouville connection

Abstract

Quantum Sturm-Liouville problems introduced in our paper (Büyükaşık et al. in J Math Phys 50:072102, 2009) provide a reach set of exactly solvable quantum damped parametric oscillator models. Based on these results, in the present paper we study a set of variable parametric nonlinear Madelung fluid models and corresponding complex Burgers equations, related to the classical orthogonal polynomials of Hermite, Laguerre and Jacobi types. We show that the nonlinear systems admit direct linearazation in the form of Schrödinger equation for a parametric harmonic oscillator, allowing us to solve exactly the initial value problems for these equations by the linear quantum Sturm-Liouville problem. For each type of equations, dynamics of the probability density and corresponding zeros, as well as the complex velocity field and related pole singularities are studied in details.

On polynomial solutions to Fokker–Planck and sinked density evolution equations

“…For r(x) = 0 the eigenfunctions generally have the same polynomial form, but are multiplied by the diffusion coefficient s(x) raised to some power, κ, specified in Sec. More recently the Laguerre polynomials were applied as solutions to the non-linear Madelung fluid equation [7] and a Burger's equation with a time-dependent forcing term [53]. Both [8] and [18] highlight the connection between the Laguerre polynomials and the algebra su(1, 1), which is then exploited to construct the coherent Laguerre function, and explore squeezed states in the Calogero-Sutherland model.…”