Soliton-like Solutions for Nonlinear Schroedinger Equation with Variable Quadratic Hamiltonians

Abstract

We construct one soliton solutions for the nonlinear Schroedinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of a complete (super) integrability of generalized harmonic oscillators. The soliton wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schroedinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painleve transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons, and Jacobi elliptic and second Painleve transcendental solutions for several variable Hamiltonians that are important for current research in nonlinear optics and Bose-Einstein condensation. The Feshbach resonance matter wave soliton management is briefly discussed from this new perspective.

22 pages, 1 figure

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Cited by in corpus (8)

• The Degenerate Parametric Oscillator and Ince's Equation
• The Minimum-Uncertainty Squeezed States for for Atoms and Photons in a Cavity
• On Integrability of Nonautonomous Nonlinear Schroedinger Equations
• Dynamical Invariants and Berry's Phase for Generalized Driven Harmonic Oscillators
• On a Hidden Symmetry of Quantum Harmonic Oscillators
• Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schrödinger equation
• Exact Wave Functions for Generalized Harmonic Oscillators
• Wave Packets and Coherent Structures for Nonlinear Schroedinger equations in Variable Nonuniform Media