A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions

Chen, Xiangjun; Chen, Zhide; Huang, Nian-Ning

doi:10.1088/0305-4470/31/33/005

Cited by 52 publications

(92 citation statements)

References 16 publications

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“…However, one still has an analytical description of the excitations around the stationary profile (51) because the expression of the excitation around a soliton is known (it is given by the squared Jost functions of the inverse problem [49], see also Appendix A of Ref. [50]).…”

Section: Discussionmentioning

confidence: 99%

Anderson localization of elementary excitations in a one-dimensional Bose-Einstein condensate

Bilas

Pavloff

2006

Eur. Phys. J. D

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Abstract. We study the elementary excitations of a transversely confined Bose-Einstein condensate in presence of a weak axial random potential. We determine the localization length (i) in the hydrodynamical low energy regime, for a domain of linear densities ranging from the Tonks-Girardeau to the transverse Thomas-Fermi regime, in the case of a white noise potential and (ii) for all the range of energies, in the "one-dimensional mean field regime", in the case where the randomness is induced by a series of randomly placed point-like impurities. We discuss our results in view of recent experiments in elongated BEC systems.

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Section: Discussionmentioning

confidence: 99%

Anderson localization of elementary excitations in a one-dimensional Bose-Einstein condensate

Bilas

Pavloff

2006

Eur. Phys. J. D

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“…Our results for t 0 and s 0 are plotted in figure 7b. As mentioned earlier, previous attempts using IST are not adequate (Chen et al 1998;Lashkin 2004;Ao & Yan 2005). t 0 was not obtained in Kivshar & Yang (1994); furthermore, no previous work has considered the evolution of the parameter s 0 .…”

Section: Linear Dampingmentioning

confidence: 95%

Perturbations of dark solitons

et al. 2011

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A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations. Keywords: perturbation theory; solitons; opticsPerturbation theory as applied to solitons that decay at infinity, i.e. so-called bright solitons, has been developed over many years (cf. Karpman & Maslov 1977;Kodama & Ablowitz 1981;Herman 1990). The analytical work employs a diverse set of methods including perturbations of the inverse scattering transform (IST), multi-scale perturbation analysis, perturbations of conserved quantities, etc.; the analysis applies to a wide range of physical problems. In optics, a central equation that describes the envelope of a quasi-monochromatic wave train is the nonlinear Schrödinger (NLS) equation, which in normalized form readswhere D, n are constants. In this paper, we consider the NLS equation in a typical nonlinear optics context, where D corresponds to the group-velocity dispersion (GVD), n > 0 is related to the nonlinear index of refraction, z is the

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“…On the other hand, the integrable defocusing NLS model supports dark solitons which are intensity dips sitting on a continuous wave background (cwb) with a phase change across their intensity minimum. In order to apply to the dark soliton perturbation and the non-vanishing boundary conditions (nvbc), inherent to this type of solution, various improvements have been made to the calculations and methods [10][11][12] based on the so-called complete set of squared Jost solution (eigenstates of the linearised NLS operator). The implementation of direct methods, however, consider small perturbations around the integrable NLS model [8,13].…”

Section: Jhep03(2016)005mentioning

confidence: 99%

Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons

Blas

Zambrano

2016

J. High Energ. Phys.

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The concept of quasi-integrability has been examined in the context of deformations of the defocusing non-linear Schrödinger model (NLS). Our results show that the quasi-integrability concept, recently discussed in the context of deformations of the sineGordon, Bullough-Dodd and focusing NLS models, holds for the modified defocusing NLS model with dark soliton solutions and it exhibits the new feature of an infinite sequence of alternating conserved and asymptotically conserved charges. For the special case of two dark soliton solutions, where the field components are eigenstates of a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved in the scattering process of the solitons. Such results are obtained through analytical and numerical methods, and employ adaptations of algebraic techniques used in integrable field theories. We perform extensive numerical simulations and consider the scattering of dark solitons for the cubic-quintic NLS model with potential V = ηI 2 − 6 I 3 and the saturable type potential satisfying V [I] = 2ηI − I q 1+I q , q ∈ Z Z + , with a deformation parameter ∈ IR and I = |ψ| 2 . The issue of the renormalization of the charges and anomalies, and their (quasi)conservation laws are properly addressed. The saturable NLS supports elastic scattering of two soliton solutions for a wide range of values of {η, , q}. Our results may find potential applications in several areas of non-linear science, such as the Bose-Einstein condensation.

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A direct perturbation theory for dark solitons based on a complete set of the squared Jost solutions

Cited by 52 publications

References 16 publications

Anderson localization of elementary excitations in a one-dimensional Bose-Einstein condensate

Anderson localization of elementary excitations in a one-dimensional Bose-Einstein condensate

Perturbations of dark solitons

Quasi-integrability in the modified defocusing non-linear Schrödinger model and dark solitons

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