Global dynamics of nonlinear dispersive equations
Introductory Workshop: Randomness and long time dynamics in nonlinear evolution differential equations August 24, 2015 - August 28, 2015
Location: MSRI: Simons Auditorium
NLS equation with potential
radially symmetric solution
existence and uniqueness results
35Q55 - NLS-like equations (nonlinear Schrödinger) [See also 37K10]
37L50 - Noncompact semigroups; dispersive equations; perturbations of Hamiltonian systems
35A01 - Existence problems: global existence, local existence, non-existence
35A02 - Uniqueness problems: global uniqueness, local uniqueness, non-uniqueness
34A12 - Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions
Solutions of nonlinear dispersive equations exhibit various space-time behavior, such as blow-up, soliton, and scattering, due to competition between the dispersion and the nonlinearity. Drastic changes are also possible along the evolution. It is hence an important and challenging problem to predict the behavior in all the future and the past from the initial data. Combining variational, dispersive, and spectral analysis, it has become possible to describe the structure of solutions and of initial data in some simple cases. In this talk I will mostly focus on the nonlinear Schrodinger equations with or without potential, which have stable and/or unstable solitons. The main goal is to obtain a global dynamical picture which contains deformation between different types of solitons
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