Journal ArticleDOI
The Stochastic Nonlinear Schrödinger Equation in H 1
A. de Bouard,Arnaud Debussche +1 more
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In this article, the local and global existence of solutions in the energy space H 1(R n ) for stochastic nonlinear Schrodinger equations with either additive or multiplicative noise was investigated.Abstract:
We investigate the local and global existence of solutions in the energy space H 1(R n ) for stochastic nonlinear Schrodinger equations with either additive or multiplicative noise. The noise is assumed to be white in time and correlated in the space variables.read more
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Stochastic Equations in Infinite Dimensions
Giuseppe Da Prato,Jerzy Zabczyk +1 more
TL;DR: In this paper, the existence and uniqueness of nonlinear equations with additive and multiplicative noise was investigated. But the authors focused on the uniqueness of solutions and not on the properties of solutions.
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Weak and Strong Order of Convergence of a Semidiscrete Scheme for the Stochastic Nonlinear Schrodinger Equation
Anne de Bouard,Arnaud Debussche +1 more
TL;DR: In this paper, the error of a semidiscrete scheme for the stochastic nonlinear Schrodinger equation with power nonlinearity was analyzed and it was shown that the numerical scheme has strong order 12$ in general and order 1 if the noise is additive.
Journal ArticleDOI
Blow-up for the stochastic nonlinear Schrödinger equation with multiplicative noise
Anne de Bouard,Arnaud Debussche +1 more
TL;DR: In this article, the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrodinger equation was studied.
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Randomly forced CGL equation: stationary measures and the inviscid limit
TL;DR: In this article, a complex Ginzburg-Landau (CGL) equation perturbed by a random force which is white in time and smooth in the space variable x is studied.
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The nonlinear Schrödinger equation with white noise dispersion
Anne de Bouard,Arnaud Debussche +1 more
TL;DR: In this paper, it was shown that the nonlinear Schrodinger equation with random dispersion converges to the non-linear non-Schrodinger with white noise dispersion in L 2 or H 1.
References
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Semilinear Schrodinger Equations
TL;DR: In this article, the linear Schrodinger equation and local Cauchy problem are studied in the repulsive case and the attractive case, respectively, and the smoothing effect is considered.
BookDOI
The nonlinear Schrödinger equation : self-focusing and wave collapse
TL;DR: In this article, the authors present a basic framework to understand structural properties and long-time behavior of standing wave solutions and their relationship to a mean field generation and acoustic wave coupling.
Journal ArticleDOI
On stochastic convolution in banach spaces and applications
TL;DR: In this paper, the authors studied stochastic evolution equations in M-type 2 Banach spaces framework using factorization method and Burkholder inequality and proved the existence of local and global solutions with close to optimal regularity.
Journal ArticleDOI
A Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise
A. de Bouard,Arnaud Debussche +1 more
TL;DR: In this paper, the existence and uniqueness of square integrable solutions for sub-critical nonlinear Schrodinger equations with a multiplicative noise was studied, and the critical exponent was shown to be the same, in dimension 1 or 2, as that of the deterministic equation.
Book
Nonlinear Random Waves
Vladimir V. Konotop,Luis Vázquez +1 more
TL;DR: In this article, Klein-Gordon models with Dynamical and Envelope solitons were used to simulate linear random waves in non-linear Media Dynamics of Randomly Modulated Solitons Waves in Non-linear Stationary Inhomogeneous Media Numerical Study of the Single-Paftiele Motion.