Access the full text.
Sign up today, get DeepDyve free for 14 days.
A. Bouard, A. Debussche (2007)
Random modulation of solitons for the stochastic Korteweg–de Vries equationAnnales De L Institut Henri Poincare-analyse Non Lineaire, 24
A. Debussche, E. Gautier (2006)
Small noise asymptotic of the timing jitter in soliton transmissionAnnals of Applied Probability, 18
T. Benjamin, J. Bona, J. Mahony (1972)
Model equations for long waves in nonlinear dispersive systemsPhilosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 272
H. Brezis (2010)
Functional Analysis, Sobolev Spaces and Partial Differential Equations
A. Debussche, Y. Tsutsumi (2010)
1D quintic nonlinear Schr\"odinger equation with white noise dispersionarXiv: Analysis of PDEs
Simon Lyons (2011)
Introduction to stochastic differential equations
A. Debussche, J. Printems (1999)
Numerical simulation of the stochastic Korteweg-de Vries equationPhysica D: Nonlinear Phenomena, 134
J. Albert (1986)
Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equationJournal of Differential Equations, 63
W. Strauss (1974)
Dispersion of low-energy waves for two conservative equationsArchive for Rational Mechanics and Analysis, 55
B Oksendal (2000)
Stochastic Differential Equations. An Introduction with Applications
Peter Kloeden (1955)
Stochastic differential equationsMathematical Proceedings of the Cambridge Philosophical Society, 51
Khaled Dika (2004)
Comportement qualitatif des solutions de l'équation de Benjamin-Bona-Mahony déterministe et stochastique
G. Lord, C. Powell, T. Shardlow (2014)
An Introduction to Computational Stochastic PDEs
A. Bouard, E. Gautier (2008)
Exit Problems Related to the Persistence of Solitons for the Korteweg-de Vries Equation with Small NoiseDiscrete and Continuous Dynamical Systems, 26
J Albert (1989)
On the decay of solutions of the generalized Benjamin–Bona–Mahony equationsJ. Math. Anal. Appl., 141
A. Bouard, W. Craig, Oliver D'iaz-Espinosa, P. Guyenne, C. Sulem (2007)
Long wave expansions for water waves over random topographyNonlinearity, 21
Tosio Kato, G. Ponce (1988)
Commutator estimates and the euler and navier‐stokes equationsCommunications on Pure and Applied Mathematics, 41
A Bouard, A Debussche (2007)
Random modulation of solitons for the stochastic Korteweg–de Vries equationAnn. IHP Nonlinear Anal., 24
R. Marty (2006)
On a splitting scheme for the nonlinear Schrödinger equation in a random mediumCommunications in Mathematical Sciences, 4
Min Chen (2003)
Equations for bi-directional waves over an uneven bottomMath. Comput. Simul., 62
K. Chouk, M. Gubinelli (2013)
Nonlinear PDEs with Modulated Dispersion I: Nonlinear Schrödinger EquationsCommunications in Partial Differential Equations, 40
A. Debussche, J. Vovelle (2013)
Invariant measure of scalar first-order conservation laws with stochastic forcingProbability Theory and Related Fields, 163
Linghai Zhang (1994)
Decay of solutions of generalized Benjamin-Bona-Mahony equationsActa Mathematica Sinica, 10
A Bouard, E Gautier (2010)
Exit problems related to the persistence of solitons for the Korteweg–de Vries equation with small noiseDCDS A, 26
P. Antonelli, J. Saut, Christof Sparber (2012)
Well-Posedness and averaging of NLS with time-periodic dispersion managementAdvances in Differential Equations
Radoin Belaouar, A. Bouard, A. Debussche (2010)
Numerical analysis of the nonlinear Schrödinger equation with white noise dispersionStochastic Partial Differential Equations: Analysis and Computations, 3
A Bouard, A Debussche (2010)
The nonlinear Schrödinger equation with white noise dispersionJ. Funct. Anal., 259
Daniel Revuz, M. Yor (1990)
Continuous martingales and Brownian motion
A. Debussche, L. Menza (2002)
Numerical simulation of focusing stochastic nonlinear Schrödinger equationsPhysica D: Nonlinear Phenomena, 162
In this article, we address the generalized BBM equation with white noise dispersion which reads $$\begin{aligned} du-du_{xx}+u_x \circ dW+ u^pu_xdt=0, \end{aligned}$$ d u - d u x x + u x ∘ d W + u p u x d t = 0 , in the Stratonovich formulation, where W(t) is a standard real valued Brownian motion. We first investigate the well-posedness of the initial value problem for this equation. We then prove theoretically and numerically that for a deterministic initial data, the expectation of the $$L^\infty _x$$ L x ∞ norm of the solutions decays to zero at $$O(t^{-\frac{1}{6}})$$ O ( t - 1 6 ) as t approaches to $$+\infty $$ + ∞ , by assuming that $$p>8$$ p > 8 and that the initial data is small in $$L^1_x\cap H^4_x$$ L x 1 ∩ H x 4 . This decay rate matches the one for solutions of the linear equation with white noise dispersion.
Stochastical Partial Differential Equations – Springer Journals
Published: Jan 9, 2017
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.