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Abstract

JEAN-MARC DELORTThis paper is devoted to the construction of periodic solutions of nonlinear Schrödinger equations on the torus, for a large set of frequencies. Usual proofs of such results rely on the use of Nash-Moser methods. Our approach avoids this, exploiting the possibility of reducing, through paradifferential conjugation, the equation under study to an equivalent form for which periodic solutions may be constructed by a classical iteration scheme. IntroductionThis paper is devoted to the existence of families of periodic solutions of Hamiltonian nonlinear Schrödinger equations on the torus ‫ޔ‬ d . Our goal is to show that such results may be proved without using Nash-Moser methods, replacing them by a technically simpler conjugation idea.We consider equations of type.!t; x; u; N u; / C f .!t; x/;where t 2 ‫,ޒ‬ x 2 ‫ޔ‬ d , F is a smooth function, vanishing at order 3 at .u; N u/ D 0, f is a smooth function on ‫ޒ‬ ‫ޔ‬ d , 2 -periodic in time, ! a frequency parameter, a real parameter and > 0 a small number. One wants to show that for small and ! in a Cantor set whose complement has small measure, the equation has time periodic solutions.Let us recall known results for that type of problems. The first periodic solutions for nonlinear wave or Schrödinger equations were constructed in [Kuksin 1993; Wayne 1990], which deal with one space dimension, with x staying in a compact interval, and imposing on the extremities of this interval convenient boundary conditions. Later on, Craig and Wayne [1993; treated the same problem for time-periodic solutions defined on ‫ޒ‬ ‫ޓ‬ 1 . Periodic solutions of nonlinear wave equations in higher space dimensions (on ‫ޒ‬ ‫ޔ‬ d , d 2) were obtained in [Bourgain 1994]. These results concern nonlinearities which are analytic. More recently, some work has been devoted to the same problem when the nonlinearity is a smooth function: Berti and Bolle [2010] have proved in this setting existence of time-periodic solutions for the nonlinear wave equation on ‫ޒ‬ ‫ޔ‬ d . We refer also to the paper of Berti, Bolle and Procesi , where the case of equations on Zoll manifolds is treated. Very recently, Berti and Procesi [2011] have studied the same problem, for wave or Schrödinger equations, on a homogeneous space. We refer also to [Craig 2000; Kuksin 2000] for more references.The proofs of all these results rely on the use of the Nash-Moser theorem, to overcome unavoidable losses of derivatives coming from the small divisors appearing when inverting the linear part of the equation. Our goal here is to show that one may construct periodic solutions of nonlinear Schrödinger equations (for large sets of frequencies), using just a standard iterative scheme instead of the quadratic scheme of the Nash-Moser method. This approach allows one to separate on the one hand the treatment of losses of derivatives coming from small divisors, and on the other hand the question of convergence of the sequence of approximations, while in a Nash-Moser scheme, both problems have to be treated at the same time....