Abstract
We consider the cubic nonlinear Schrödinger equation posed on the spatial domain$\mathbb{R}\times \mathbb{T}^{d}$. We prove modified scattering and construct modified wave operators for small initial and final data respectively ($1\leqslant d\leqslant 4$). The key novelty comes from the fact that the modified asymptotic dynamics are dictated by theresonant systemof this equation, which sustains interesting dynamics when$d\geqslant 2$. As a consequence, we obtain global strong solutions (for$d\geqslant 2$) with infinitely growing high Sobolev norms $H^{s}$.
Highlights
The purpose of this work is to study the asymptotic behavior of the cubic defocusing nonlinear Schrodinger (NLS) equation posed on the waveguide manifolds R × Td: (i ∂t + ∆)U = |U |2U, (1.1)
We want to understand how this asymptotic behavior is related to a resonant dynamic, in a case when scattering does not occur
The other interesting feature of the asymptotic dynamics of (1.1) as opposed to previous modified scattering results is that the modification dictated by its resonant system is not a phase correction term when d 2, but rather a much more vigorous departure from linear dynamics. This will pose a new set of difficulties in comparison to previous modified scattering results in the literature, but, on the plus side, will lead us to several interesting and new types of asymptotic dynamics
Summary
The purpose of this work is to study the asymptotic behavior of the cubic defocusing nonlinear Schrodinger (NLS) equation posed on the waveguide manifolds R × Td:. The best thing is to find the simplest possible dynamical system that describes the asymptotic dynamics of F(t) To find this system, one has to work on proving global a priori energy and decay estimates that allow one to decompose the nonlinearity in the F equation in the following way:. Previous modified scattering results that we are aware of only concerned equations (or systems) posed on Rd, quasilinear or semilinear [1, 25, 32, 33, 53,54,55, 63, 64, 66, 74, 89], and had an integrable asymptotic system for (1.6) This often allowed for a simple phase conjugation (in physical or Fourier space) to give the modification.
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