Discretization effects in the nonlinear Schrödinger equation

Fibich, Gadi; Ilan, Boaz

doi:10.1016/s0168-9274(02)00112-5

Cited by 18 publications

(21 citation statements)

References 23 publications

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“…In our NLS simulations we use standard fourth-order finite-difference schemes for the spatial derivatives and explicit fourth-order Runge-Kutta for marching in z. As has recently been shown in [9], in finite-difference simulations of NLS solutions that are known analytically to become singular, the computed solution still remains bounded. Therefore, there is always an element of arbitrariness in selecting a numerical criterion for blowup in NLS simulations.…”

Section: Resultsmentioning

confidence: 99%

“…The difference between the two is due to backscattering and can be used to quantify the level of backscattering for a particular setting; see [10,12]. 9 In Table 4.2 we provide the values of maximum self-focusing and maximum backscattering in the NLH, defined as max r,z |E(z, r)| and max r |E(0, r) − E 0 inc (r)|, respectively, for various values of and δ. The dash "-" in a particular cell of Table 4.2 means that the level of damping was insufficient to guarantee the convergence of the numerical algorithm.…”

Section: Resultsmentioning

confidence: 99%

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Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

Ilan

Fibich

Tsynkov³

2003

SIAM J. Appl. Math.

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The critical nonlinear Schrödinger equation (NLS) models the propagation of intense laser light in Kerr media. This equation is derived from the more comprehensive nonlinear Helmholtz equation (NLH) by employing the paraxial approximation and neglecting the backscattered waves. It is known that if the input power of the laser beam (i.e., L 2 norm of the initial solution) is sufficiently high, then the NLS model predicts that the beam will self-focus to a point (i.e., collapse) at a finite propagation distance. Mathematically, this behavior corresponds to the formation of a singularity in the solution of the NLS. A key question which has been open for many years is whether the solution to the NLH, i.e., the "parent" equation, may nonetheless exist and remain regular everywhere, particularly for those initial conditions (input powers) that lead to blowup in the NLS. In the current study we address this question by introducing linear damping into both models and subsequently comparing the numerical solutions of the damped NLH (boundary-value problem) with the corresponding solutions of the damped NLS (initial-value problem) for the case of one transverse dimension. Linear damping is introduced in much the same way as is done when analyzing the classical constant-coefficient Helmholtz equation using the limiting absorption principle. Numerically, we have found that it provides a very efficient tool for controlling the solutions of both the NLH and NLS. In particular, we have been able to identify initial conditions for which the NLS solution does become singular, while the NLH solution still remains regular everywhere. We believe that our finding of a larger domain of existence for the NLH than for the NLS is accounted for by precisely those mechanisms that have been neglected when deriving the NLS from the NLH, i.e., nonparaxiality and backscattering.

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Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

Ilan

Fibich

Tsynkov³

2003

SIAM J. Appl. Math.

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“…This equation is the modified equation of the semi-discrete second-order nonlinear Schrödinger equation (see [11]). Clearly, the following equations are special cases of (1.1):…”

Section: Introductionmentioning

confidence: 99%

“…But for Eq. (1.1), both local well-posedness and global well-posedness are sparsely studied; the only available results are those of Fibich and his co-workers [10][11][12][13][14][15] on the investigation of possible critical exponents for global existence, but those results do not consider local existence. Their results are based on the conservation laws of (1.1).…”

Section: Introductionmentioning

confidence: 99%

Well-posedness of the Cauchy problem of high dimension non-isotropic fourth-order Schrödinger equations in Sobolev spaces

Guo

Cui

2009

Nonlinear Analysis: Theory, Methods & Applications

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“…This paper is concerned with the Cauchy problem of the following fourth-order nonlinear dispersive equation in R n × R: (1) iu t + ∆u + |u| α u + a d i u xixixixi = 0, x ∈ R n , t ∈ R. u(x, 0) = ϕ(x),…”

Section: Introductionmentioning

confidence: 99%

Global Small Solutions of the Cauchy Problem for Nonisotropic Schrödinger Equations

Zhao¹,

Cui²

2008

Kyungpook mathematical journal

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In this paper we study the existence of global small solutions of the Cauchy problem for the non-isotropically perturbed nonlinear Schrödinger equation: iut + ∆u + |u| α u + a d i ux i x i x i x i = 0, where a is real constant, 1 ≤ d < n is a integer, α is a positive constant, and x = (x1, x2, • • • , xn) ∈ R n. For some admissible α we show the existence of global(almost global) solutions and we calculate the regularity of those solutions.

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Discretization effects in the nonlinear Schrödinger equation

Cited by 18 publications

References 23 publications

Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

Backscattering and Nonparaxiality Arrest Collapse of Damped Nonlinear Waves

Well-posedness of the Cauchy problem of high dimension non-isotropic fourth-order Schrödinger equations in Sobolev spaces

Global Small Solutions of the Cauchy Problem for Nonisotropic Schrödinger Equations

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