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Quasi-periodic solutions of nonlinear Schrödinger equations on \(T^{d}\). (English) Zbl 1230.35126

Authors’ abstract: We present recent existence results of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on \(\mathbb{T}^d\), \(d\geq 1\), finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are in \(C^\infty\) then the solutions are in \(C^\infty\). The proofs are based on an improved Nash-Moser iterative scheme and a new multiscale inductive analysis for the inverse linearized operators.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K55 Perturbations, KAM theory for infinite-dimensional Hamiltonian and Lagrangian systems
37K50 Bifurcation problems for infinite-dimensional Hamiltonian and Lagrangian systems
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