Numerical study of two‐dimensional stochastic NLS equations
Copyright policy )Numerical study of two‐dimensional stochastic NLS equations
AbstractIn 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.
- École Normale Supérieure de Rennes France
- École des Ponts ParisTech France
- Mathematics Research Institute of Rennes France
- Universite Rennes 2 France
- University of Rennes 2 France
Numerical solutions to stochastic differential and integral equations, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], nonlinear Schrödinger equations, numerical examples, stochastic partial differential equations, NLS equations (nonlinear Schrödinger equations), nonlinear Shrödinger equations, refinement procedure, 65M06, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 510, multiplicative and additive noise, finite difference schemes, 35Q55, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 76B25, Stochastic partial differential equations (aspects of stochastic analysis), Finite difference methods for initial value and initial-boundary value problems involving PDEs, 60H15, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Computational methods for stochastic equations (aspects of stochastic analysis)
Numerical solutions to stochastic differential and integral equations, [MATH.MATH-PR] Mathematics [math]/Probability [math.PR], nonlinear Schrödinger equations, numerical examples, stochastic partial differential equations, NLS equations (nonlinear Schrödinger equations), nonlinear Shrödinger equations, refinement procedure, 65M06, [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], 510, multiplicative and additive noise, finite difference schemes, 35Q55, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], 76B25, Stochastic partial differential equations (aspects of stochastic analysis), Finite difference methods for initial value and initial-boundary value problems involving PDEs, 60H15, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Computational methods for stochastic equations (aspects of stochastic analysis)
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