Abstract
In this paper we continue some investigations on the periodic NLSEiu u +iu xx +u|u|p-2 (p≦6) started in [LRS]. We prove that the equation is globally wellposed for a set of data Φ of full normalized Gibbs measrue\(e^{ - \beta H(\phi )} Hd\phi (x),H(\phi ) = \tfrac{1}{2}\int {\left| {\phi '} \right|^2 - \tfrac{1}{p}\int {\left| \phi \right|p} } \) (after suitableL 2-truncation). The set and the measure are invariant under the flow. The proof of a similar result for the KdV and modified KdV equations is outlined. The main ingredients used are some estimates from [B1] on periodic NLS and KdV type equations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
[Bid] Bidegaray, B.: Mesures invariantes pour des équations aux dérivées partielles. Preprint Orsay
[B1] Bourgain, J.: Fourier restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Geom. and Funt. Anat.3, No 2, 107–156, 209–262 (1993)
[B2] Bourgain, J.: On the longtime behaviour of nonlinear Hamiltonian evolution equations. Preprint IHES 4/94, to appear in Geom. and Funct. Anal. (GAFA)
[B3] Bourgain, J.: Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Preprint IHES 4/94
[G-V] Ginibre, J., Velo, G.: Ann. Inst. H. Poincaré28, 287–316 (1978)
[MCK-V] Mckean, H., Vaninski, K.: Statistical mechanics of nonlinear wave equations and brownian motion with restoring drift: The petit and micro-canonical ensembless. Commun. Math. Phys.160, 615–630 (1994) and preprint 1993
[ML-S] MacLaughlin, D., Schober, C.: Chaotic and homolinic behavior for numerical discretization of the nonlinear Schrödinger equation. Physica D57, 447–465 (1992)
[L-R-S] Lebowitz, J., Rose, R., Speer E.: Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys. V50, 657–687 (1988)
[Zhl] Zhidkov, P.: On the invariant measure for the nonlinear Schrödinger equation. Doklady Akad Nauk SSSR317, No 3, 543 (1991)
[Zh2] Zhidkov, P.: An invariant measure for a nonlinear wave equation. Preprint 1992
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bourgain, J. Periodic nonlinear Schrödinger equation and invariant measures. Commun.Math. Phys. 166, 1–26 (1994). https://doi.org/10.1007/BF02099299
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02099299