Explore global dynamics in nonlinear dispersive partial differential equations, focusing on solitons in this 39-minute lecture by Kenji Nakanishi for the International Mathematical Union. Delve into the nonlinear Klein-Gordon equation, examining solutions slightly above ground state energy and in the energy space. Investigate stable and unstable solitons, using the nonlinear Schrödinger equation as a model for two-soliton systems. Learn about mass-energy restrictions and global dynamics with stable/unstable solitons. Analyze two-soliton interactions in the nonlinear Klein-Gordon equation with constant damping, discussing soliton instability. Conclude by considering open questions regarding three-soliton systems and soliton mergers, providing insights into cutting-edge research in nonlinear dynamics.
Kenji Nakanishi - Global Dynamics Around and Away From Solitons
International Mathematical Union via YouTube
Overview
Syllabus
Intro
Nonlinear dispersive PDEs, such as the nonlinear Klein-Gordon equation
Slightly above the ground state energy Consider the nonlinear Klein-Gordon (KG) in the energy space
Stable and unstable solitons To consider solution sets containing two distinct solitons in size, shape and stability , an easy model is the nonlinear Schrodinger equation
Mass-energy restriction
Global dynamics with stable/unstable solitons (N. '17)
2-solitons To deal with global dynamics containing multi-solitons, we consider the nonlinear Klein-Gordion with a constant o damping
Instability of the solitons
Open questions: 3-soliton and soliton merger
Open question: soliton merger
Taught by
International Mathematical Union
