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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

Citations:

Zbl 1142.35090
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References:

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