Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

Bao, Weizhu

doi:10.4310/maa.2004.v11.n3.a8

Cited by 30 publications

(48 citation statements)

References 24 publications

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“…spectral finite difference (TSCP) and the time-splitting finite difference with transformation (TSFD-T) methods [16,17] will be made in order to control the highly oscillatory phase background. In fact, these methods allowed us to improve in several orders of magnitude the accuracy in the computation of the charges and anomalies presented in [4].…”

Section: Jhep03(2016)005mentioning

confidence: 99%

“…the boundary condition (4.1) is satisfied for each time step. In our numerical simulations we will use the so-called time-splitting cosine pseudo-spectral finite difference (TSCP) and the time-splitting finite difference with transformation (TSFD-T) methods [16,17] for k = 0 and k = 0, respectively. Our numerical simulations reproduce the main properties already known for dark soliton interactions in the integrable defocusing NLS model.…”

Section: Jhep03(2016)005mentioning

confidence: 99%

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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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Section: Jhep03(2016)005mentioning

confidence: 99%

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 proposed numerical method is similar to the one used to study the dynamics of rotating BEC [3,4], where the angular momentum rotation term becomes a constant with the adoption of polar coordinates. The extensive numerical results presented in this article demonstrate that the method is very efficient and accurate, and when applied to study interaction and dynamics of vortex lattices in the GLSE (1.1), it is capable of producing conclusive simulation results on the vortex stability and dynamic properties.…”

Section: N(ψ(· T)) =mentioning

confidence: 99%

Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation

Zhang

Bao

2007

Eur. J. Appl. Math

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The rich dynamics of quantized vortices governed by the Ginzburg-Landau-Schrödinger equation (GLSE) is an interesting problem studied in many application fields. Although recent mathematical analysis and numerical simulations have led to a much better understanding of such dynamics, many important questions remain open. In this article, we consider numerical simulations of the GLSE in two dimensions with non-zero far-field conditions. Using twodimensional polar coordinates, transversely highly oscillating far-field conditions can be efficiently resolved in the phase space, thus giving rise to an unconditionally stable, efficient and accurate time-splitting method for the problem under consideration. This method is also time reversible for the case of the non-linear Schrödinger equation. By applying this numerical method to the GLSE, we obtain some conclusive experimental findings on issues such as the stability of quantized vortex, interaction of two vortices, dynamics of the quantized vortex lattice and the motion of vortex with an inhomogeneous external potential. Discussions on these simulation results and the recent theoretical studies are made to provide further understanding of the vortex stability and vortex dynamics described by the GLSE.

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“…Numerical and analytical results suggest that the vortex states with winding number m = ±1 are dynamically stable, and, respectively, |m| > 1 dynamically unstable [27,34,25,26,22,2,36] (note that the stability and interaction laws of a quantized vortex in the Gross-Pitaevskii equation for BEC [3,4,5] may be very different from that studied here due to the different far-field boundary conditions).…”

Section: Introductionmentioning

confidence: 97%

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

Zhang¹,

Bao²,

Du³

2007

SIAM J. Appl. Math.

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Abstract. The dynamic laws of quantized vortex interactions in the Ginzburg-Landau-Schrödinger equation (GLSE) are analytically and numerically studied. A review of the reduced dynamic laws governing the motion of vortex centers in the GLSE is provided. The reduced dynamic laws are solved analytically for some special initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently in [Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., to appear], we can qualitatively and quantitatively compare quantized vortex interaction patterns of the GLSE with those from the reduced dynamic laws. Some conclusive findings are obtained, and discussions on numerical and theoretical results are made to provide further understanding of vortex interactions in the GLSE. Finally, the vortex motion under an inhomogeneous potential in the GLSE is also studied.

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Numerical Methods for the Nonlinear Schrödinger Equation with Nonzero Far-field Conditions

Abstract: Abstract. In this paper we present numerical methods for the nonlinear Schrödinger equations (NLS) in the semiclassical regimes:

Cited by 30 publications

References 24 publications

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

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

Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation

The Dynamics and Interaction of Quantized Vortices in the Ginzburg–Landau–Schrödinger Equation

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