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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
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