Analytical solutions and integrable structure of the time-dependent harmonic oscillator with friction

Hongli AN, Wai Hong CHAN, Biao LI, Man Wai YUEN

Research output: Contribution to journalArticle


By employing the Madelung transformation, the time-dependent harmonic oscillator with friction described by the Schrödinger equation is reduced to a hydrodynamic system. An exponential elliptic vortex ansatz is introduced, and thereby a finite-dimensional nonlinear dynamical system is obtained. Time-modulated physical variables corresponding to the divergence, spin, shear, and normal deformation rates of the Madelung velocity field are introduced, and the dynamical system is reducible to a form amenable to general solutions. In particular, three typical elliptical vortex solutions termed pulsrodons are derived, and their behaviours are simulated. These solutions have recently found applications in oceanic and atmospheric dynamics. Moreover, it is shown that the harmonic oscillator with friction has an underlying integrable structure of Ermakov–Hamiltonian type. Copyright © 2019 Walter de Gruyter GmbH, Berlin/Boston.
Original languageEnglish
JournalZeitschrift fur Naturforschung - Section A Journal of Physical Sciences
Early online dateFeb 2019
Publication statusE-pub ahead of print - Feb 2019


Harmonic Oscillator
dynamical systems
harmonic oscillators
Analytical Solution
Vortex flow
Nonlinear dynamical systems
Nonlinear Dynamical Systems
General Solution
Velocity Field
Dynamical systems
velocity distribution
Dynamical system

Bibliographical note

An, H., Chan, W., Li, B., & Yuen, M. (2019). Analytical solutions and integrable structure of the time-dependent harmonic oscillator with friction. Zeitschrift für Naturforschung A. Advance online publication. doi: 10.1515/zna-2018-0412


  • Analytical solutions
  • Elliptic vortex ansatz
  • Ermakov–Hamiltonian systems
  • Integrable structure
  • Time-dependent harmonic oscillator