Numerical simulation of focusing stochastic nonlinear Schrödinger equations

Debussche, Arnaud; Menza, Laurent Di

doi:10.1016/s0167-2789(01)00379-7

Cited by 98 publications

(102 citation statements)

References 20 publications

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“…This is related to the results of [8] where it is proved that for every positive T , P(T (u) < T ) > 0 and to the graphs in Section 4 of [10].…”

Section: Proof Since T Is Lower Semicontinuous T −1 ((T +∞]) Is Anmentioning

confidence: 69%

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Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Gautier

2005

ESAIM: PS

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Abstract. Sample path large deviations for the laws of the solutions of stochastic nonlinearSchrödinger equations when the noise converges to zero are presented. The noise is a complex additive Gaussian noise. It is white in time and colored in space. The solutions may be global or blow-up in finite time, the two cases are distinguished. The results are stated in trajectory spaces endowed with topologies analogue to projective limit topologies. In this setting, the support of the law of the solution is also characterized. As a consequence, results on the law of the blow-up time and asymptotics when the noise converges to zero are obtained. An application to the transmission of solitary waves in fiber optics is also given.Mathematics Subject Classification. 35Q51, 35Q55, 60F10, 60H15.

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“…This is related to the results of [8] where it is proved that for every positive T , P(T (u) < T ) > 0 and to the graphs in Section 4 of [10].…”

Section: Proof Since T Is Lower Semicontinuous T −1 ((T +∞]) Is Anmentioning

confidence: 69%

“…It corresponds to the approximation of the blow-up time used in [10]. We obtain the following bounds.…”

Section: Bounds For the Approximate Blow-up Timementioning

confidence: 96%

Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Gautier

2005

ESAIM: PS

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“…Note that, when σ is an integer, f (a, b) is a polynomial in the variables a 2 and b 2 and is easy to evaluate precisely. In [3], [12], finite differences have been used for the spatial discretization and a refinement procedure has been used to capture blow-up solutions in the presence of noise. Note that the discretization of the noise corresponds to a Ito product in (2.1).…”

Section: Preliminariesmentioning

confidence: 99%

“…Numerical methods are often used in order to understand the behaviour of the solutions. Numerical simulations have been performed for example in [2,3,12] to understand the effect of the noise on blow-up phenomena.…”

Section: Introductionmentioning

confidence: 99%

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Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation

2006

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Abstract.In this article, we analyze the error of a semi discrete scheme for the stochastic non linear Schrödinger equation with power nonlinearity. We consider supercritical or subcritical nonlinearity and the equation can be either focusing of defocusing. Allowing sufficient spatial regularity we prove that the numerical scheme has strong order 1/2 in general and order 1 if the noise is additive. Furthermore, we also prove that the weak order is always 1.

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Numerical study of two‐dimensional stochastic NLS equations

Barton-Smith¹,

Debussche²,

Menza³

2005

Numerical Methods Partial

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In this article, we numerically solve the two‐dimensional stochastic nonlinear Schrödinger equation in the case of multiplicative and additive white noises. The aim is to investigate their influence on well‐known deterministic solutions: stationary states and blowing‐up solutions. In the first case, we find that a multiplicative noise has a damping effect very similar to diffusion. However, for small amplitudes of the noise, the structure of solitary state is still localized. In the second case, a local refinement algorithm is used to overcome the difficulty arising for the computation of singular solutions. Our experiments show that multiplicative white noise stops the deterministic blow‐up that occurs in the critical case. This extends the results of Debussche and Di Menza (Physica D, 162(3–4) 2002, 131–154) in the one‐dimensional case. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential, 2005.

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Numerical simulation of focusing stochastic nonlinear Schrödinger equations

Cited by 98 publications

References 20 publications

Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Large deviations and support results for nonlinear Schrödinger equations with additive noise and applications

Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation

Numerical study of two‐dimensional stochastic NLS equations

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