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

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Journal of Russian Laser Research Aims and scope

Abstract

We construct one-soliton solutions for the nonlinear Schr¨odinger 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 Schr¨odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev´e 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 Painlev´e 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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Puerto Rico, Mayagüez, call box 9000, PR 00681-9000, Puerto Rico

    Erwin Suazo

  2. School of Mathematical & Statistical Sciences and Mathematical & Computational Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1804, U.S.A.

    Sergei K. Suslov

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  1. Erwin Suazo
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  2. Sergei K. Suslov
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Correspondence to Sergei K. Suslov.

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Suazo, E., Suslov, S.K. Soliton-like solutions for the nonlinear schrödinger equation with variable quadratic hamiltonians. J Russ Laser Res 33, 63–83 (2012). https://doi.org/10.1007/s10946-012-9261-3

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  • DOI: https://doi.org/10.1007/s10946-012-9261-3

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