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SUMMARY:Classification of Initial Data for Global Dynamics of Nonlinear Di
 spersive Equations
DTSTART:20260527T120000Z
DTEND:20260527T140000Z
DTSTAMP:20260422T234000Z
UID:indico-event-15177@indico.math.cnrs.fr
CONTACT:cecile@ihes.fr
DESCRIPTION:Speakers: Kenji Nakanishi (RIMS Kyoto)\n\nNonlinear dispersive
  equations are partial differential equations to describe various wave phe
 nomena where the primary effects are wave dispersion and nonlinear interac
 tions. Even a single equation can have many different types of solutions d
 epending on the initial data\, such as scattering\, blow-up\, and solitons
 .\nThe theme of this course is to classify global behavior of solutions in
  terms of the initial data. More precisely\, the problem is to characteriz
 e the set of initial data corresponding to each type of solutions\, togeth
 er with the configuration of those sets\, which also requires to analyze t
 ransient evolutions during intermediate time. Despite the recent progress 
 for the soliton resolution conjecture\, which classifies the asymptotic be
 havior\, its link to the initial data is much less understood\, mostly res
 tricted in the data size\, types of behavior\, and by symmetry of the equa
 tion or the solutions.\nThe lecture will focus on two model cases as attem
 pts to extend it in two directions. The first is to extend the initial dat
 a set to more variety of solutions\; we consider the nonlinear Klein-Gordo
 n equation and initial data near superposition of the ground state soliton
 s\, which are unstable. It is natural to expect that the classification is
  also a superposition of the single soliton case\, but the interactions am
 ong unstable modes of different growth rates and large radiation from coll
 apsed solitons can possibly spoil such a simple picture\, by energy transf
 er 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 we
 ighted energy tailored for radiations from multi-solitons.\nThe 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 Hamilt
 onian and mass conservation\, but without the Galilei or Lorentz invarianc
 e\, nor the center of mass or energy. Such loss of structure poses serious
  difficulty especially in proving the rigidity that the minimal non-disper
 sive solutions must be the ground states. I will show how to overcome it\,
  by combining virial-variational estimates and space-time estimates for no
 n-radiative source terms.\n\nhttps://indico.math.cnrs.fr/event/15177/
LOCATION:Amphithéâtre Léon Motchane (IHES)
URL:https://indico.math.cnrs.fr/event/15177/
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