"BLOW UP, DISPERSION AND SOLITONS"

Objectif

"Many physical models involve nonlinear dispersive problems, like wave
or laser propagation, plasmas, ferromagnetism, etc. So far, the mathematical under-
standing of these equations is rather poor. In particular, we know little about the
detailed qualitative behavior of their solutions. Our point is that an apparent com-
plexity hides universal properties of these models; investigating and uncovering such
properties has started only recently. More than the equations themselves, these univer-
sal properties are essential for physical modelisation.
By considering several standard models such as the nonlinear Schrodinger, nonlinear
wave, generalized KdV equations and related geometric problems, the goal of this pro-
posal is to describe the generic global behavior of the solutions and the proﬁles which
emerge either for large time or by concentration due to strong nonlinear eﬀects, if pos-
sible through a few relevant solutions (sometimes explicit solutions, like solitons). In
order to do this, we have to elaborate diﬀerent mathematical tools depending on the
context and the speciﬁcity of the problems. Particular emphasis will be placed on
- large time asymptotics for global solutions, decomposition of generic solutions into
sums of decoupled solitons in non integrable situations,
- description of critical phenomenon for blow up in the Hamiltonian situation, stable
or generic behavior for blow up on critical dynamics, various relevant regularisations of
the problem,
- global existence for defocusing supercritical problems and blow up dynamics in the
focusing cases.
We believe that the PI and his team have the ability to tackle these problems at present.
The proposal will open whole ﬁelds of investigation in Partial Diﬀerential Equations in
the future, clarify and simplify our knowledge on the dynamical behavior of solutions
of these problems and provide Physicists some new insight on these models."

Champ scientifique

Régime de financement

ERC-AG - ERC Advanced Grant

Institution d’accueil

UNIVERSITE DE CERGY-PONTOISE

Contribution de l’UE

€ 2 079 798,00

Adresse

BOULEVARD DU PORT 33
95011 Cergy-Pontoise
France

Type d’activité

Higher or Secondary Education Establishments

Contact administratif

Laurence Puechberty (Ms.)

Chercheur principal

Franck Merle (Prof.)

Liens

Contacter l’organisation Site web

Coût total

Aucune donnée

Bénéficiaires (1)