Journal of Mathematical Chemistry

, Volume 50, Issue 10, pp 2716–2745

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

Original Paper

DOI: 10.1007/s10910-012-0060-4

Cite this article as:
Büyükaşık, Ş.A. & Pashaev, O.K. J Math Chem (2012) 50: 2716. doi:10.1007/s10910-012-0060-4

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.

Keywords

Schroedinger equation Damped parametric harmonic oscillator Sturm–Liouville problems Quantum hydrodynamics Madelung fluid Burgers equation Pole dynamics Time variable parameters Exact solvability 

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsIzmir Institute of TechnologyUrla, IzmirTurkey

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