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.


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
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