Modified scattering and beating effect for coupled Schrödinger systems on product spaces with small initial data

Abstract

In this paper, we study a coupled nonlinear Schrödinger system with small initial data in a product space. We establish a modified scattering of the solutions of this system and we construct a modified wave operator. The study of the resonant system, which provides the asymptotic dynamics, allows us to highlight a control of the Sobolev norms and interesting dynamics with the beating effect. The proof uses a recent work of Hani, Pausader, Tzvetkov, and Visciglia for the modified scattering, and a recent work of Grébert, Paturel, and Thomann for the study of the resonant system.

Growth of Sobolev norms for coupled lowest Landau level equations

“…It is likely that one can obtain polynomial bounds on the Sobolev norms for (1.18) using the ideas of Sohinger [36,37]. A modified scattering result was obtained in [41], and the existence of unbounded orbits (on a wave guide) in the case σ = 1 follows from [26]. In [23] a non-linear phenomenon was exhibited on (1.18) posed on T. Moreover, let us mention that traveling waves solutions (in H 1 ) exist for Schrödinger equations on R d and for coupled Schrödinger systems as well : see in particular [29,16] where traveling waves with different speeds are constructed.…”

Growth of Sobolev norms for coupled Lowest Landau Level equations

“…It is likely that one can obtain polynomial bounds on the Sobolev norms for (1.18) using the ideas of Sohinger [27,28]. A modified scattering result was obtained in [31], and the existence of unbounded orbits (on a wave guide) in the case σ = 1 follows from [19]. In [16] a non-linear phenomenon was exhibited on (1.18) posed on T. Moreover, let us mention that traveling waves solutions exist for Schrödinger equations on R d with zero or nonzero conditions at infinity (see [9] and references therein), but such solutions do not infer growth of Sobolev norms in this setting.…”

Resonant Hamiltonian systems associated to the one-dimensional nonlinear Schrödinger equation with harmonic trapping

“…Resonant equations have been used to construct solutions of the cubic NLS on T 2 that exhibit large growth of Sobolev norms [10]. They have appeared as modified scattering limits for a number of equations, including the cubic NLS on R × T d [21], the cubic NLS on R d with 2 ≤ d ≤ 5 and harmonic trapping in all but one direction [22], and a coupled cubic NLS system on R × T [26]. The continuous resonant equation (CR) was originally shown to approximate the dynamics of small solutions of the twodimensional cubic NLS on a large torus T 2 L over long times scales (longer than L 2 /ǫ 2 , where ǫ is the size of the initial data) [12].…”