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Résumé |
Nonlinear dispersive equations are partial differential equations to describe various wave phenomena where the primary effects are wave dispersion and nonlinear interactions. Even a single equation can have many different types of solutions depending on the initial data, such as scattering, blow-up, and solitons.
The theme of this course is to classify global behavior of solutions in terms of the initial data. More precisely, the problem is to characterize the set of initial data corresponding to each type of solutions, together with the configuration of those sets, which also requires to analyze transient evolutions during intermediate time. Despite the recent progress for the soliton resolution conjecture, which classifies the asymptotic behavior, its link to the initial data is much less understood, mostly restricted in the data size, types of behavior, and by symmetry of the equation or the solutions.
The lecture will focus on two model cases as attempts to extend it in two directions. The first is to extend the initial data set to more variety of solutions; we consider the nonlinear Klein-Gordon equation and initial data near superposition of the ground state solitons, which are unstable. It is natural to expect that the classification is also a superposition of the single soliton case, but the interactions among unstable modes of different growth rates and large radiation from collapsed solitons can possibly spoil such a simple picture, by energy transfer from the most unstable mode to the others. I will show how to preclude it by using elementary geometry of the Lorentz transform and space-time weighted energy tailored for radiations from multi-solitons.
The second is to extend the equations to less symmetry; we consider the Zakharov system, which is a system of the Schrodinger and the wave equations with Hamiltonian and mass conservation, but without the Galilei or Lorentz invariance, nor the center of mass or energy. Such loss of structure poses serious difficulty especially in proving the rigidity that the minimal non-dispersive solutions must be the ground states. I will show how to overcome it, by combining virial-variational estimates and space-time estimates for non-radiative source terms. |
