SOLITON-LIKE SOLUTIONS FOR THE NONLINEAR SCHRODINGER EQUATION WITH VARIABLE QUADRATIC HAMILTONIANS

作者：Suazo Erwin; Suslov Sergei K^*

来源：Journal of Russian Laser Research, 2012, 33(1): 63-83.

• nonlinear Schrodinger equation
• Gross-Pitaevskii equation
• Bose-Einstein condensation
• Feshbach resonance
• fiber optics
• generalized harmonic oscillators
• soliton-like solutions
• Jacobian elliptic functions
• Painleve II transcendents

We construct one-soliton solutions for the nonlinear Schrodinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the 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 Schrodinger 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 research in nonlinear optics, plasma physics, and Bose-Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.

• 出版日期2012-1