## Invariant measures for the defocusing nonlinear Schrödinger equation.(English)Zbl 1171.35116

The paper is concerned with the nonlinear Schrödinger equation $(i\partial_t+\Delta)u-F(u)=0\quad\text{in }\mathbb{R}\times\Theta \tag{NLS}$ on the open unit disc $$\Theta\subset\mathbb{R}^2$$ with Dirichlet boundary conditions. The nonlinearity is of the form $$F(z)=G'(|z|^2)z$$ with $$G:\mathbb{R}\to\mathbb{R}$$ smooth, and it is defocusing and subquintic. A model example is $$F(z)=|z|^\alpha z$$ with $$0<\alpha<4$$. The author proves the existence of a Gibbs type measure, on a suitable phase space, associated to the radial solutions of (NLS) which is invariant under the global flow of (NLS). The result extends an earlier paper [Dyn. Partial Differ. Equ. 3, No. 2, 111–160 (2006; Zbl 1142.35090)] where the author considered subcubic, focusing or defocusing nonlinearities.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) 37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems 81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Zbl 1142.35090
Full Text:

### References:

 [1] Anton, R., Cubic nonlinear Schrödinger equation on three dimensional balls with radial data, (2006) [2] Ayache, A.; Tzvetkov, N.$$, L^p$$ properties of Gaussian random series · Zbl 1145.60019 [3] Bourgain, J., Periodic nonlinear Schrödinger equation and invariant measures, Comm. Math. Phys., 166, 1-26, (1994) · Zbl 0822.35126 [4] Bourgain, J., Invariant measures for the 2D-defocusing nonlinear Schrödinger equation, Comm. Math. Phys., 176, 421-445, (1996) · Zbl 0852.35131 [5] Burq, N.; Gérard, P.; Tzvetkov, N., Zonal low regularity solutions of the nonlinear Schrödinger equation on $$S^d, (2002)$$ · Zbl 1003.35113 [6] Burq, N.; Gérard, P.; Tzvetkov, N., Multilinear eigenfunction estimates and global existence for the three dimensional nonlinear Schrödinger equations, Ann. ENS, 38, 255-301, (2005) · Zbl 1116.35109 [7] Christ, M.; Colliander, J.; Tao, T., Ill-posedness for nonlinear Schrödinger and wave equations, (2003) [8] Ginibre, J., Le problème de Cauchy pour des EDP semi-linéaires périodiques en variables d’espace (d’après Bourgain), Séminaire Bourbaki, Exp. 796, Astérisque, 237, 163-187, (1996) · Zbl 0870.35096 [9] Kuksin, S.; Shirikyan, A., Randomly forced CGL equation : stationary measures and the inviscid limit, J. Phys A, 37, 1-18, (2004) · Zbl 1047.35061 [10] Lebowitz, J.; Rose, R.; Speer, E., Statistical dynamics of the nonlinear Schrödinger equation, J. Stat. Physics V, 50, 657-687, (1988) · Zbl 1084.82506 [11] Stein, E.; Weiss, G.; Princeton University Press, Princeton N. J., Introduction to Fourier analysis on Euclidean spaces, 32, (1971) · Zbl 0232.42007 [12] Tzvetkov, N., Invariant measures for the nonlinear Schrödinger equation on the disc, Dynamics of PDE, 3, 111-160, (2006) · Zbl 1142.35090 [13] Zhidkov, P., Korteweg de Vries and nonlinear Schrödinger equations : qualitative theory, 1756, (2001), Springer-Verlag, Berlin · Zbl 0987.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.