Exact Solutions to the Nonlinear Schrödinger Equation

Aktosun, Tuncay; Busse, Theresa; Demontis, Francesco; Mee, Cornelis van der

doi:10.1007/978-3-0346-0161-0_1

Cited by 6 publications

(7 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the present paper we complement the previous comparisons by answering which of the time invariants (3), (4), in addition to mass (2), is most important to conserve and how efficiently this can be done. A further aim is to find competitive methods to use in a low regularity regime and for d > 1.…”

Section: Introductionmentioning

confidence: 82%

“…A more difficult test case related to signal propagation in optical fibers and involving two interacting stationary solitons was constructed in [4]. In this example we are looking for the solution u of the following NLSE with a focusing nonlinearity…”

Section: Two Stationary Solitonsmentioning

confidence: 99%

“…Using a finite element based spatial discretization, we test these five mass-conservative methods on problems of increasing difficulty in one and two dimensions. The test cases include solitons with known analytical solution such as the single traveling soliton and a newly derived stationary soliton [4]. Problems of physical relevance involving optical lattices and angular momentum are also considered.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Henning¹,

Wärnegård²

2019

Kinetic &Amp; Related Models

View full text Add to dashboard Cite

In this paper we present a numerical comparison of various mass-conservative discretizations for the time-dependent Gross-Pitaevskii equation. We have three main objectives. First, we want to clarify how purely mass-conservative methods perform compared to methods that are additionally energy-conservative or symplectic. Second, we shall compare the accuracy of energy-conservative and symplectic methods among each other. Third, we will investigate if a linearized energy-conserving method suffers from a loss of accuracy compared to an approach which requires to solve a full nonlinear problem in each time-step. In order to obtain a representative comparison, our numerical experiments cover different physically relevant test cases, such as traveling solitons, stationary multi-solitons, Bose-Einstein condensates in an optical lattice and vortex pattern in a rapidly rotating superfluid. We shall also consider a computationally severe test case involving a pseudo Mott insulator. Our space discretization is based on finite elements throughout the paper. We will also give special attention to long time behavior and possible coupling conditions between time-step sizes and mesh sizes. The main observation of this paper is that mass conservation alone will not lead to a competitive method in complex settings. Furthermore, energy-conserving and symplectic methods are both reliable and accurate, yet, the energy-conservative schemes achieve a visibly higher accuracy in our test cases. Finally, the scheme that performs best throughout our experiments is an energy-conserving relaxation scheme with linear time-stepping proposed by C. Besse (SINUM,42(3):934-952,2004).

show abstract

Section: Introductionmentioning

confidence: 82%

Section: Two Stationary Solitonsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Henning¹,

Wärnegård²

2019

Kinetic &Amp; Related Models

View full text Add to dashboard Cite

show abstract

“…The experiment considers the case of two stationary solitons that are interacting we each other and it was first described in [4] and numerically studied in [32]. The combined behavior of the two solitons is characterized as the solution u to the following focusing Gross-Pitaevskii equation with cubic nonlinearity,…”

Section: A Numerical Benchmark Problemmentioning

confidence: 99%

Superconvergence of time invariants for the Gross-Pitaevskii equation

Henning¹,

Wärnegård²

2020

Preprint

View full text Add to dashboard Cite

This paper considers the numerical treatment of the time-dependent Gross-Pitaevskii equation. In order to conserve the time invariants of the equation as accurately as possible, we propose a Crank-Nicolson-type time discretization that is combined with a suitable generalized finite element discretization in space. The space discretization is based on the technique of Localized Orthogonal Decompositions (LOD) and allows to capture the time invariants with an accuracy of order O(H 6 ) with respect to the chosen mesh size H. This accuracy is preserved due to the conservation properties of the time stepping method. Furthermore, we prove that the resulting scheme approximates the exact solution in the L ∞ (L 2 )-norm with order O(τ 2 + H 4 ), where τ denotes the step size. The computational efficiency of the method is demonstrated in numerical experiments for a benchmark problem with known exact solution.

show abstract

“…An appearance of the latter in a Riemann-Hilbert factorization problem dates back to 1986 [27]. Sylvester's equation typically enters the stage when matrix (or operator) versions of integrable (differential or difference) equations are considered [28], or when matrix methods are applied, e.g., to concisely express iterated Bäcklund or Darboux transformations (see, e.g., [29] and references therein, and [30,31]). It is therefore not surprising that, for various integrable equations, different specializations of Sylvester's equation show up via the universal binary Darboux transformation in the framework of bidifferential calculus (see [23,24] and references cited there).…”

Section: )mentioning

confidence: 99%

NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks

Chvartatskyi¹,

Müller‐Hoissen²

2017

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

The infinite families of Peregrine, Akhmediev and Kuznetsov-Ma breather solutions of the focusing Nonlinear Schrödinger (NLS) equation are obtained via a matrix version of the Darboux transformation, with a spectral matrix of the form of a Jordan block. The structure of these solutions is essentially determined by the corresponding solution of the Lyapunov equation. In particular, regularity follows from properties of the Lyapunov equation.

show abstract

Exact Solutions to the Nonlinear Schrödinger Equation

Cited by 6 publications

References 12 publications

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation

Superconvergence of time invariants for the Gross-Pitaevskii equation

NLS breathers, rogue waves, and solutions of the Lyapunov equation for Jordan blocks

Contact Info

Product

Resources

About