arXiv subjects

Raphaël Cote

Publications and source records attributed to Raphaël Cote.

2 recordsLinked to original sources

arXiv2020

Polynomial Growth of high Sobolev Norms of solutions to the Zakharov-Kuznetsov Equation

We consider the Zakharov-Kuznetsov equation (ZK) in space dimension 2. Solutions u with initial data u\_0 $\in$ H s are known to be global if s $\ge$ 1. We prove that for any integer s $\ge$ 2, u(t) H s grows at most polynomially in t for large times t. This result is related to wave turbulence and how a solution of (ZK) can move energy to high frequencies. It is inspired by analoguous results by Staffilani on the non linear Schr{ö}dinger Korteweg-de-Vries equation. The main ingredients are adequate bilinear estimates in the context of Bourgain's spaces.

math.AP↗

arXiv2011

High speed excited multi-solitons in nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with a general nonlinearity. In dimension higher than 2, this equation admits travelling wave solutions with a fixed profile which is not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable.

math.AP↗