The nonlinear Schrödinger equation on tori: Integrating harmonic analysis, geometry, and probability

Abstract

Abstract. The field of nonlinear dispersive and wave equations has undergone significant progress in the last twenty years thanks to the influx of tools and ideas from nonlinear Fourier and harmonic analysis, geometry, analytic number theory and most recently probability, into the existing functional analytic methods. In these lectures we concentrate on the semilinear Schrödinger equation defined on tori and discuss the most important developments in the analysis of these equations. In particular, we discuss in some detail recent work by J. Bourgain and C. Demeter proving the 2 decoupling conjecture and as a consequence the full range of Strichartz estimates on either rational or irrational tori, thus settling an important earlier conjecture by Bourgain.

Nonexistence of small, smooth, time-periodic, spatially periodic solutions for nonlinear Schrödinger equations

“…We then define W δ = W n+1,δ . Lemma 4 also guarantees the existence of a constant d 1 > 0 for which (13) holds, for all T ∈ W δ .…”

“…Another motivation is the difficulty in understanding dynamics of nonlinear dispersive equations on the torus. (We refer the interested reader to [13] for a recent survey of the study of nonlinear Schrödinger equations on the torus.) On free space, quite a large number of scattering results have been proven; we mention just [16] as an example.…”

The orthonormal Strichartz inequality on torus

“…Moreover, it was also observed that the inequality (1.2) still holds replacing the torus by more general irrational torus. For further discussion and the theory on the irrational torus including survey, see [11,15,21,24,29,33,36]. It is notable that in [6], Burq-Gérard-Tzvetkov studied the nonlinear Schrödinger equation on the compact manifold.…”

Construction of solutions for some localized nonlinear Schrödinger equations

“…ϕ has exactly m zeroes, ϕ = 0 only at isolated points. 2can also be adapted to study non-linear Schrödinger equations on the one dimensional torus (related aspects can be found in [32]). Namely, consider the equation…”