On scattering for the defocusing quintic nonlinear Schrödinger equation on the two-dimensional cylinder

Abstract

In this article, we prove the scattering for the quintic defocusing nonlinear Schrödinger equation on cylinder R × T in H 1 . We establish an abstract linear profile decomposition in L 2x h α , 0 < α ≤ 1, motivated by the linear profile decomposition of the mass-critical Schrödinger equation in L 2 (R d ), d ≥ 1. Then by using the solution of the one-discrete-component quintic resonant nonlinear Schrödinger system, whose scattering can be proved by using the techniques in 1d mass critical NLS problem by B. Dodson, to approximate the nonlinear profile, we can prove scattering in H 1 by using the concentration-compactness/rigidity method. As a byproduct of our proof of the scattering of the one-discrete-component quintic resonant nonlinear Schrödinger system, we also prove the conjecture of the global well-posedness and scattering of the two-discrete-component quintic resonant nonlinear Schrödinger system made by Z. Hani and B. Pausader [Comm. Pure Appl. Math. 67 (2014)].

Scattering for the mass super-critical perturbations of the mass critical nonlinear Schrödinger equations

“…Then as in [10], we can directly apply the linear profile decomposition of the Schrödinger equations in L 2 , but keep our eyes open that the initial data is in H 1 , we can exclude one direction of the scaling limit. After making the contradiction that the solution does not scatter, we can find a sequence of solutions u n : R × R d → C with u n ∈ A ω,+ , and…”

On Global-in-Time Strichartz Estimates for the Semiperiodic Schrödinger Equation

“…We will assume throughout the section that f = P ≤N f . Let M (f ) denote the mixed norm on the left-hand side of (5). In order to show that M (f ) ǫ N ǫ f L 2 , we decouple the frequencies to reduce to the case where f has small Fourier support.…”

“…Now (7) shows that in order to prove (5) we can in fact assume that the Fourier transform of f is supported in a cube of side length ∼ 1 in the frequency space R n × Z d . Let P θ f denote a smooth frequency cut-off of f onto θ.…”

“…In this paper we focus on the setting of product manifolds of the form R n × T d , where T d is a (rational or irrational) d-dimensional torus. There has been recent interest in the behavior of solutions to the linear and nonlinear Schrödinger equation on these manifolds (see for example [5], [8], [9], [11], [12], [19]). In particular, one can exploit dispersive effects coming from the Euclidean component of the manifold to obtain stronger asymptotic results than in the setting of T d .…”

Global Well-posedness for the Focusing Cubic NLS on the Product Space $\mathbb{R} \times \mathbb{T}^3$

“…The techniques used in Euclidean and tori settings are frequently combined and applied to the waveguides problems. We refer to [8,9,10,16,19,20,21,22,33,35,34,36] for some NLS results in the waveguide setting.…”