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"BLOW UP, DISPERSION AND SOLITONS"

Final Report Summary - BLOWDISOL (BLOW UP, DISPERSION AND SOLITONS)

This project deals with the mathematical analysis of several universal models of Physics such as the nonlinear Schrodinger equation, the nonlinear wave equation, the generalized Korteweg-de Vries equation and related geometric problems. We focus on models where no special integrability structure is available and where the development of robust tools is essential for future research. The main goal 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 possible through a few relevant solutions, like solitons (solutions which do not change shape in time). An important problem related to such questions is to identify and understand the process of simplification (loss of information in time) in the context of reversible dynamics.

Particular emphasis is placed on:
- The large time asymptotics for all or generic global solutions. The objective is to prove that, even in non-integrable situations, asymptotically in time, solutions decompose into sums of decoupled solitons plus a linear radiation. This remarkable phenomenon is known as the soliton resolution conjecture.
- The phenomenon of formation of singularities (blow up) for Hamiltonian models with critical nonlinearity. The description should include the concentration rate and the blow up profile of the solution. Critical here means that the nonlinear effects are in critical balance with linear effects.
- The formation of singularity in energy supercritical focusing problems. For supercritical problems, the nonlinear effects can be strictly stronger than linear effects.

At the completion of the project, the BLOWDISOL team has obtained among others the following breakthrough results:
- First proof of the soliton resolution conjecture in a non-integrable situation, by a completely new approach. To obtain this result, the authors have successfully implemented a road map with several new concepts and have unlocked the way to a new set of problems with possible high risk/high gain targets in the field of mathematical analysis. This has opened a new direction of research in Mathematical Physics.
- The proof of the inelastic character of solitons collision for any parameter for the energy critical wave equation.
- The complete dynamical classification of the behavior of solutions around solitons for the mass critical generalized Korteweg-de Vries equation. This is the first complete result of this nature in critical dispersive situations.
- The construction of new almost explicit blow up solutions for energy supercritical problems. For the nonlinear Schrodinger equation, we have extended to supercritical problems a methodology known only for critical problems. For the supercritical heat equation, we have obtained new blow up rates and new blow up profiles from fully non isotropic constructions. From these works, a new set of problems can be investigated.
- Existence of blow up solutions for the Schrodinger maps problem. This is a highly technical achievement in the context of critical problems.