We prove that a system of coupled nonlinear Schrödinger equations on the torus exhibits both stable and unstable small KAM tori. In particular the unstable tori are related to a beating phenomena which has been proved recently in [6]. This is the first example of unstable tori for a 1d PDE.

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.

Riemann’s non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch–Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely, the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.

The purpose of this note is to propose a study of various nonlinear behaviors for a system of two coupled cubic Schrödinger equations with small initial data. Depending on the choice of the spatial domain, we highlight different examples of nonlinear behaviors. The goal is to mix the approaches of the study on the torus (with a truly nonlinear behavior) and of the study on the real line (with an infinite behavior) in order to obtain on the product space R × T a truly nonlinear behavior in infinite time.

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