## Periodic nonlinear Schrödinger equation and invariant measures.(English)Zbl 0822.35126

Summary: We continue some investigations on the periodic NLSE $iu_ t+ u_{xx}+ u| u|^{p-2} =0 \qquad (p\leq 6)$ started in [J. Lebowitz, R. Rose and E. Speer, J. Stat. Phys. 50, 657-687 (1988)]. We prove that the equation is globally wellposed for a set of data $$\varphi$$ of full normalized Gibbs measure $e^{- \beta H(\varphi)} Hd\varphi (x), \qquad H(\varphi)= {\textstyle {1\over 2}}\int |\varphi' |^ 2- {\textstyle {1\over p}} \int| \varphi|^ p$ (after suitable $$L^ 2$$-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 the author [Geom. Funct. Anal. 3, No. 2, 107-156, No. 3, 209-262 (1993; Zbl 0787.35097 and Zbl 0787.35098)] on periodic NLS and KdV type equations.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 37C10 Dynamics induced by flows and semiflows

### Citations:

Zbl 0787.35097; Zbl 0787.35098
Full Text:

### References:

 [1] [Bid] Bidegaray, B.: Mesures invariantes pour des équations aux dérivées partielles. Preprint Orsay [2] [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) · Zbl 0787.35097 [3] [B2] Bourgain, J.: On the longtime behaviour of nonlinear Hamiltonian evolution equations. Preprint IHES 4/94, to appear in Geom. and Funct. Anal. (GAFA) [4] [B3] Bourgain, J.: Invariant measures for the 2D-defocusing nonlinear Schrödinger equation. Preprint IHES 4/94 · Zbl 0852.35131 [5] [G-V] Ginibre, J., Velo, G.: Ann. Inst. H. Poincaré28, 287–316 (1978) [6] [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 · Zbl 0792.60077 [7] [ML-S] MacLaughlin, D., Schober, C.: Chaotic and homolinic behavior for numerical discretization of the nonlinear Schrödinger equation. Physica D57, 447–465 (1992) · Zbl 0760.35045 [8] [L-R-S] Lebowitz, J., Rose, R., Speer E.: Statistical mechanics of the nonlinear Schrödinger equation. J. Stat. Phys. V50, 657–687 (1988) · Zbl 1084.82506 [9] [Zhl] Zhidkov, P.: On the invariant measure for the nonlinear Schrödinger equation. Doklady Akad Nauk SSSR317, No 3, 543 (1991) [10] [Zh2] Zhidkov, P.: An invariant measure for a nonlinear wave equation. Preprint 1992
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