Ground states for the higher-order dispersion managed NLS equation in the absence of average dispersion

Kunze, Markus; Moeser, Jamison T.; Zharnitsky, Vadim

doi:10.1016/j.jde.2004.09.014

Cited by 13 publications

(9 citation statements)

References 16 publications

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“…going back to [6]. For a simple explicit proof of (B.9), see for example, [31] or [24]. Now assume that the supports of f l and f m have positive distance for some l, m ∈ {1, 2, 3, 4}.…”

Section: And Only If It Converges Weakly and Limmentioning

confidence: 99%

On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

Hundertmark

Lee

2011

J Nonlinear Sci

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We give a simple proof of existence of solutions of the dispersion management and diffraction management equations for zero average dispersion, respectively diffraction. These solutions are found as maximizers of non-linear and non-local variational problems which are invariant under a large non-compact group. Our proof of existence of maximizer is rather direct and avoids the use of Lions' concentration compactness argument or Ekeland's variational principle.

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“…going back to [6]. For a simple explicit proof of (B.9), see for example, [31] or [24]. Now assume that the supports of f l and f m have positive distance for some l, m ∈ {1, 2, 3, 4}.…”

Section: And Only If It Converges Weakly and Limmentioning

confidence: 99%

On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

Hundertmark

Lee

2011

J Nonlinear Sci

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“…Similar definition holds in the periodic case where R n is replaced by T n with T = [0, 2π[ with periodic boundary conditions. The (randomly) modulated NLS equation has been subject of interest in recent literature (for example [1,16,17,37,38,40,45,51]), especially related to applications to soliton management in optical wave-guides. We were directly inspired by the recent work of De Bouard and Debussche [16] who study the Nonlinear Schrödinger equation with Brownian modulation.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger Equations

Chouk

Gubinelli

2015

Communications in Partial Differential Equations

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We start a study of various nonlinear PDEs under the effect of a modulation in time of the dispersive term. In particular in this paper we consider the modulated non-linear Schrödinger equation (NLS) in dimension 1 and 2 and the derivative NLS in dimension 1. We introduce a deterministic notion of "irregularity" for the modulation and obtain local and global results similar to those valid without modulation. In some situations, we show how the irregularity of the modulation improves the well-posedness theory of the equations. We develop two different approaches to the analysis of the effects of the modulation. A first approach is based on novel estimates for the regularising effect of the modulated dispersion on the non-linear term using the theory of controlled paths. A second approach is an extension of a Strichartz estimated first obtained by Debussche and Tsutsumi in the case of the Brownian modulation for the quintic NLS.

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“…In the last decades, the modulated Schrödinger equation (1) has gained a lot of attention in physics serving, e.g., as a model describing propagation of light waves in optical dispersionmanaged fibers (see for instance [1,7,10,11,20,21,25,32]). Numerically, however, only very little is known so far for this type of problem.…”

Section: Introductionmentioning

confidence: 99%

“…Convergence plot of the Strang splitting scheme(25), the classical exponential integrator(24) and the randomized exponential integrator (11) with 100 sequences in the case of a smooth function g (left) and a non-regular, 1/2-Hölder continuous function g (right). Hölder continuous function.…”

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confidence: 99%

Randomized exponential integrators for modulated nonlinear Schrödinger equations

Hofmanová

Knöller

Schratz

2020

IMA Journal of Numerical Analysis

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We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha ,2}$ for some $\alpha \in (0,1)$. Due to the loss of smoothness in the problem, classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha +1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods.

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Ground states for the higher-order dispersion managed NLS equation in the absence of average dispersion

Cited by 13 publications

References 16 publications

On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

On Non-local Variational Problems with Lack of Compactness Related to Non-linear Optics

Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger Equations

Randomized exponential integrators for modulated nonlinear Schrödinger equations

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