arXiv subjects

Victor Vilaça da Rocha

Publications and source records attributed to Victor Vilaça da Rocha.

4 recordsLinked to original sources

arXiv2021

Intermittency of Riemann's non-differentiable function through the fourth-order flatness

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.

math.CA↗

arXiv2018

Stable and unstable time quasi periodic solutions for a system of coupled NLS equations

We prove that a system of coupled nonlinear Schr{ö}dinger 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.

math.AP↗

arXiv2017

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

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.

math.AP↗

arXiv2015

Asymptotic behavior of solutions to the cubic coupled Schrödinger systems in one space dimension

In this paper, we study a coupled nonlinear Schr{ö}dinger system with small initial data in the one dimension Euclidean space. Such a system appears in the context of the coupling between two different optical waveguides. We establish an asymptotic nonlinear behavior and a decay estimate for solutions of this system. The proof uses a recent work of Kato and Pusateri.

math.AP↗