Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations

Eilbeck, J. C.; Enolskii, V. Z.; Kostov, N. A.

doi:10.1063/1.1318733

Cited by 38 publications

(22 citation statements)

References 38 publications

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“…Further perspectives of finding stable periodic solutions to the n-component case are outlined in [16]. For n = 2 finitegap solutions of Manakov system given in terms of multidimensional functions are derived in [24,25]; reduction of finite-gap solutions to elliptic functions is presented in [13].…”

Section: Discussionmentioning

confidence: 99%

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Two-component Bose-Einstein condensates in periodic potential

et al. 2004

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Coupled nonlinear Schrödinger (CNLS) equations with an external elliptic function potential model with high accuracy a quasi-one-dimensional interacting two-component Bose-Einstein condensate (BEC) trapped in a standing wave generated by a few laser beams. The construction of stationary solutions of the two-component CNLS equation with a periodic potential is detailed and their stability properties are studied by direct numerical simulations. Some of these solutions allow reduction to the Manakov system. From a physical point of view the trivial phase solutions can be interpreted as exact Bloch states at the edge of the Brillouin zone. Some of them are stable while others are found to be unstable against weak modulations of long wavelength. By numerical simulations it is shown that the modulationally unstable solutions lead to the formation of localized ground states of the coupled BEC system.

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Section: Discussionmentioning

confidence: 99%

“…Such solutions for the one-component NLS equation are well known; see [13] and the numerous references therein. Elliptic solutions for the CNLS equation and Manakov system were derived in [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Two-component Bose-Einstein condensates in periodic potential

et al. 2004

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“…The detailed study of two-phase solutions of the focusing NLS equation (1) is important for understanding the small-dispersion limit near the gradient catastrophe [9] and, in particular, the generation of rogue waves near the gradient catastrophe [10]. Special classes of ultra-elliptic solutions of vector NLS equations have also been of much interest recently [11,12,13,14], including their modulation equations [15]. For elliptic solutions of the NLS equation, Kamchatnov [1,16] has demonstrated an effective integration method that clarifies the dependence of the solution on the branch points of the underlying elliptic curve.…”

Section: Introductionmentioning

confidence: 99%

Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

Wright

2016

Physica D: Nonlinear Phenomena

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An effective integration method based on the classical solution to the Jacobi inversion problem, using Kleinian ultra-elliptic functions, is presented for quasi-periodic two-phase solutions of the focusing nonlinear Schrödinger equation. The two-phase solutions with real quasi-periods are known to form a two-dimensional torus, modulo a circle of complex phase factors, that can be expressed as a ratio of theta functions. In this paper, the two-phase solutions are explicitly parametrized in terms of the branch points on the genus-two Riemann surface of the theta functions. Simple formulas, in terms of the imaginary parts of the branch points, are obtained for the maximum modulus and the minimum modulus of the two-phase solution.

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“…Such solutions for the one-component nonlinear Schrödinger equation are well known, see [20] and the numerous references therein. Elliptic solutions for the CNLS and Manakov system were derived in [21,22,23].…”

Section: Basic Equationsmentioning

confidence: 99%

Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

Kostov

Gerdjikov

Valchev

2007

SIGMA

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Abstract. We present two new families of stationary solutions for equations of Bose-Fermi mixtures with an elliptic function potential with modulus k. We also discuss particular cases when the quasiperiodic solutions become periodic ones. In the limit of a sinusoidal potential (k → 0) our solutions model a quasi-one dimensional quantum degenerate BoseFermi mixture trapped in optical lattice. In the limit k → 1 the solutions are expressed by hyperbolic function solutions (vector solitons). Thus we are able to obtain in an unified way quasi-periodic and periodic waves, and solitons. The precise conditions for existence of every class of solutions are derived. There are indications that such waves and localized objects may be observed in experiments with cold quantum degenerate gases.

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Quasiperiodic and periodic solutions for vector nonlinear Schrödinger equations

Cited by 38 publications

References 38 publications

Two-component Bose-Einstein condensates in periodic potential

Two-component Bose-Einstein condensates in periodic potential

Effective integration of ultra-elliptic solutions of the focusing nonlinear Schrödinger equation

Exact Solutions for Equations of Bose-Fermi Mixtures in One-Dimensional Optical Lattice

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