Analytical Solutions and Integrable Structure of the Time-Dependent Harmonic Oscillator With Friction

Abstract

Abstract 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.