### 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) |
---|---|

Pages (from-to) | 63-83 |

Number of pages | 21 |

Journal | Journal of Russian Laser Research |

Volume | 33 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2012 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

**Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians.** / Suazo, Erwin; Suslov, Sergei.

Research output: Contribution to journal › Article

*Journal of Russian Laser Research*, vol. 33, no. 1, pp. 63-83. https://doi.org/10.1007/s10946-012-9261-3

}

TY - JOUR

T1 - Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians

AU - Suazo, Erwin

AU - Suslov, Sergei

PY - 2012/1

Y1 - 2012/1

N2 - 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.

AB - 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.

KW - Bose-Einstein condensation

KW - Feshbach resonance

KW - fiber optics

KW - generalized harmonic oscillators

KW - Gross-Pitaevskii equation

KW - Jacobian elliptic functions

KW - nonlinear Schrödinger equation

KW - Painlevé II transcendents

KW - soliton-like solutions

UR - http://www.scopus.com/inward/record.url?scp=84858749650&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858749650&partnerID=8YFLogxK

U2 - 10.1007/s10946-012-9261-3

DO - 10.1007/s10946-012-9261-3

M3 - Article

AN - SCOPUS:84858749650

VL - 33

SP - 63

EP - 83

JO - Journal of Russian Laser Research

JF - Journal of Russian Laser Research

SN - 1071-2836

IS - 1

ER -