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"BLOW UP, DISPERSION AND SOLITONS"

Objective

"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."

Call for proposal

ERC-2011-ADG_20110209
See other projects for this call

Host institution

UNIVERSITE DE CERGY-PONTOISE
Address
Boulevard Du Port 33
95011 Cergy-pontoise
France
Activity type
Higher or Secondary Education Establishments
EU contribution
€ 2 079 798
Principal investigator
Franck Merle (Prof.)
Administrative Contact
Laurence Puechberty (Ms.)

Beneficiaries (1)

UNIVERSITE DE CERGY-PONTOISE
France
EU contribution
€ 2 079 798
Address
Boulevard Du Port 33
95011 Cergy-pontoise
Activity type
Higher or Secondary Education Establishments
Principal investigator
Franck Merle (Prof.)
Administrative Contact
Laurence Puechberty (Ms.)