Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation

Bao, Weizhu; Tang, Qinglin; Xu, Zhiguo

doi:10.1016/j.jcp.2012.10.054

Cited by 91 publications

(70 citation statements)

References 34 publications

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“…In order to simulate the time dependence of field configurations for computing bright soliton quantities we have used an efficient and accurate numerical method, the so-called time-splitting cosine pseudo-spectral finite difference method (TSCP) [9,10], in order to control the highly oscillatory phase background. This method allowed us to improve in several orders of magnitude the accuracy in the computations of the charges and anomalies presented in [5].…”

Section: Jhep05(2017)106mentioning

confidence: 99%

Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons

Blas

Bonfim

Vilela

2017

J. High Energ. Phys.

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Deformations of the focusing non-linear Schrödinger model (NLS) are considered in the context of the quasi-integrability concept. We strengthen the results of JHEP 09 (2012) 103 for bright soliton collisions. We addressed the focusing NLS as a complement to the one in JHEP 03 (2016) 005, in which the modified defocusing NLS models with dark solitons were shown to exhibit an infinite tower of exactly conserved charges. We show, by means of analytical and numerical methods, that for certain two-bright-soliton solutions, in which the modulus and phase of the complex modified NLS field exhibit even parities under a space-reflection symmetry, the first four and the sequence of even order charges are exactly conserved during the scattering process of the solitons. We perform extensive numerical simulations and consider the bright solitons with deformed potential V = 2η 2+ |ψ| 2 2+ , ∈ R, η < 0. However, for two-soliton field components without definite parity we also show numerically the vanishing of the first non-trivial anomaly and the exact conservation of the relevant charge. So, the parity symmetry seems to be a sufficient but not a necessary condition for the existence of the infinite tower of conserved charges. The model supports elastic scattering of solitons for a wide range of values of the amplitudes and velocities and the set {η, }. Since the NLS equation is ubiquitous, our results may find potential applications in several areas of non-linear science.

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Section: Jhep05(2017)106mentioning

confidence: 99%

Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons

Blas

Bonfim

Vilela

2017

J. High Energ. Phys.

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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%

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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“…As an additional example to test the MSD boundary condition, we simulate two equal-charge vortices whose interaction is known to produce a rotating circular motion of the two vortices orbiting each other [11,33,44,45]. Using a fixed grid size of 171 × 171, the simulations are run for long times using the L0 and MSD boundary conditions.…”

Section: Two-dimensional Dark Vortices In the Nlsementioning

confidence: 99%

A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schrödinger Equation

Caplan¹,

Carretero-González²

2014

SIAM J. Sci. Comput.

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Abstract. An easy to implement modulus-squared Dirichlet (MSD) boundary condition is formulated for numerical simulations of time-dependent complex partial differential equations in multidimensional settings. The MSD boundary condition approximates a constant modulus-squared value of the solution at the boundaries and is defined as 1. Introduction. When utilizing numerical methods to approximate the solutions to time-dependent partial differential equations (PDEs), proper handling of boundary conditions can be quite challenging. Sometimes, an otherwise stable numerical scheme will become unstable depending on how the boundary conditions are computed [42]. In addition, high-order schemes can degrade in accuracy to lower order when using boundary conditions which are not compatible with the high-order accuracy [26]. Proper handling of boundary conditions in higher-order schemes, especially in high-order compact schemes, can be even more of a challenge [19,20].Often, researchers will forgo a complicated boundary condition implementation and instead use tried-and-true boundary condition techniques which are very simple yet provide acceptable results. One of the most common is the use of Dirichlet boundary conditions when simulating solutions which decay towards zero at infinity, and where most of the dynamics (or "action") is expected to remain in the central regions of the computational grid. Another simple method in such cases is to use periodic boundary conditions.Infinite-domain problems involving PDEs whose function values are complex cannot, in general, make use of numerical Dirichlet or periodic boundary conditions because of the oscillation of the real and imaginary parts of the function due to the intrinsic frequency of the system. (In cases when both the real and imaginary parts of the solution converge to a constant at infinity, such boundary conditions can be

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Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation

Cited by 91 publications

References 34 publications

Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons

Quasi-integrable non-linear Schrödinger models, infinite towers of exactly conserved charges and bright solitons

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

A Modulus-Squared Dirichlet Boundary Condition for Time-Dependent Complex Partial Differential Equations and Its Application to the Nonlinear Schrödinger Equation

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