### Abstract

We construct one-soliton solutions for the nonlinear Schr̈odinger 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 Schr̈odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev́e 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 Painlev́e 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.

Original language | English (US) |
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Pages (from-to) | 63-83 |

Number of pages | 21 |

Journal | Journal of Russian Laser Research |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2012 |

### Keywords

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

### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics
- Engineering (miscellaneous)