Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

Abstract

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases where the bifurcation equation is infinitedimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.

Periodic solutions for the Schrödinger equation with nonlocal smoothing nonlinearities in higher dimension

“…Such an assumption could be removed, see [22]. For D = 2 we do not impose any further condition on f , whereas for D 3 we shall consider a more restrictive class of nonlinearities, by requiring f (x, u,ū) = ∂ ∂ū H (x, u,ū) + g(x,ū), H (x, u,ū) = H (x, u,ū), (1.4) i.e.…”

Quasiperiodic motions in dynamical systems: Review of a renormalization group approach

“…By using the tree formalism, small amplitude periodic solutions have been proved to exist, in dimension D = 1, both in the nonresonant case -µ in a suitable Cantor set [49] -and in the resonant case -µ = 0 [50,51]. Results have been obtained also in the higher space dimensional case D > 1 [52]. We refer to the original papers for a precise formulation of the results and the proofs.…”

Periodic solutions of nonlinear Schrödinger equations: a paradifferential approach

“…Related results, concerning the case of rational frequencies, may be found in [Berti 2007, Chapter 5]. Recently, Gentile and Procesi [2009] found, for analytic nonlinearities, an alternative approach to Nash-Moser using expansions in terms of Lindsted series.…”