Modified scattering for the cubic Schrödinger equation on product spaces and applications
We consider the cubic nonlinear Schrödinger equation posed on the spatial domain R× Td.
We prove modified scattering and construct modified wave operators for small initial and …
We prove modified scattering and construct modified wave operators for small initial and …
An abstract Birkhoff normal form theorem and exponential type stability of the 1d NLS
We study stability times for a family of parameter dependent nonlinear Schrödinger
equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first …
equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first …
[HTML][HTML] A KAM algorithm for the resonant non-linear Schrödinger equation
C Procesi, M Procesi - Advances in Mathematics, 2015 - Elsevier
We prove, through a KAM algorithm, the existence of large families of stable and unstable
quasi-periodic solutions for the NLS in any number of independent frequencies. The main …
quasi-periodic solutions for the NLS in any number of independent frequencies. The main …
Rational normal forms and stability of small solutions to nonlinear Schrödinger equations
J Bernier, E Faou, B Grebert - Annals of PDE, 2020 - Springer
We consider general classes of nonlinear Schrödinger equations on the circle with nontrivial
cubic part and without external parameters. We construct a new type of normal forms …
cubic part and without external parameters. We construct a new type of normal forms …
[HTML][HTML] Growth of Sobolev norms for the analytic NLS on T2
We consider the completely resonant non-linear Schrödinger equation on the two
dimensional torus with any analytic gauge invariant nonlinearity. Fix s> 1. We show the …
dimensional torus with any analytic gauge invariant nonlinearity. Fix s> 1. We show the …
On Scattering for the Quintic Defocusing Nonlinear Schrödinger Equation on R × T2
We consider the problem of large‐data scattering for the quintic nonlinear Schrödinger
equation on R× T2. This equation is critical both at the level of energy and mass. Most …
equation on R× T2. This equation is critical both at the level of energy and mass. Most …
Sobolev norms explosion for the cubic NLS on irrational tori
We consider the cubic nonlinear Schrödinger equation on 2-dimensional irrational tori. We
construct solutions which undergo growth of Sobolev norms. More concretely, for every s> 0 …
construct solutions which undergo growth of Sobolev norms. More concretely, for every s> 0 …
Birkhoff normal forms for Hamiltonian PDEs in their energy space
J Bernier, B Grébert - Journal de l'École polytechnique …, 2022 - numdam.org
We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian
partial differential equations on confined domains. Provided that the system enjoys a new …
partial differential equations on confined domains. Provided that the system enjoys a new …
Growth of Sobolev norms for the quintic NLS on T2
E Haus, M Procesi - Analysis & PDE, 2015 - msp.org
We study the quintic nonlinear Schrödinger equation on a two-dimensional torus and exhibit
orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently …
orbits whose Sobolev norms grow with time. The main point is to reduce to a sufficiently …
Long time dynamics for generalized Korteweg–de Vries and Benjamin–Ono equations
J Bernier, B Grébert - Archive for Rational Mechanics and Analysis, 2021 - Springer
We provide an accurate description of the long time dynamics of the solutions of the
generalized Korteweg–De Vries and Benjamin–Ono equations on the one dimension torus …
generalized Korteweg–De Vries and Benjamin–Ono equations on the one dimension torus …