Kenji Nakanishi - Initial classification of global behavior around multi-solitons for the nonlin... 

This talk was part of the Thematic Programme on "Nonlinear Waves and Relativity" held at the ESI April 29 -- June 21, 2024.
This is joint work with Gong Chen (GATech). We study the nonlinear Klein-Gordon equation with the focusing cubic power in three space dimensions. Our goal is to classify the initial data for the global behavior in an open neighborhood of multi-solitons generated from the ground state. The main difficulty is to trace the unstable mode of each soliton, whose exponential growth rate depends on its speed. An ODE model suggests that the most unstable mode can screw up the growth rates and directions of the slower ones, even if their interactions are indirect with exponential decay factors. It may sound like the end of story, as the dispersive estimates do not decay exponentially, but our situation turns out to be better. The key mechanism is growth delay during energy transfer between solitons, and our key tool is an energy estimate with exponential space-time weight for the linearized equation around mutli-solitons.

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