The nonlinear Schrödinger equation with white noise dispersion

Anne de Bouard (1) , Arnaud Debussche (2, 3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IPSO - Invariant Preserving SOlvers

Under certain scaling the nonlinear Schrödinger equation with random dispersion converges to the nonlinear Schrödinger equation with white noise dispersion. The aim of this work is to prove that this latter equation is globally well posed in $L^2$ or $H^1$. The main ingredient is the generalization of the classical Strichartz estimates. Additionally, we justify rigorously the formal limit described above.

Mots clés: White noise dispersion; Strichartz estimates; stochastic partial differential equation; nonlinear fiber optics

Domaines: Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Anne de Bouard, Arnaud Debussche. The nonlinear Schrödinger equation with white noise dispersion. Journal of Functional Analysis, 2010, 259 (5), pp.1300-1321. ⟨10.1016/j.jfa.2010.04.002⟩. ⟨hal-00465310⟩