Kenji Nakanishi (Kyoto University)

The APDE seminar on Monday, 9/20, will be given by Kenji Nakanishi (Kyoto University) online via Zoom from 4:10pm to 5:00pm PST. To participate, email Sung-Jin Oh (

Title: Global dynamics around 2-solitons for the nonlinear damped Klein-Gordon equation

Abstract: This is joint work with Kenjiro Ishizuka (Kyoto). We study global behavior of solutions for the nonlinear Klein-Gordon equation with a damping and a focusing nonlinearity on the Euclidean space. Recently, Cote, Martel and Yuan proved the soliton resolution conjecture for this equation completely in the one-dimensional case: every global solution in the energy space is asymptotic to superposition of solitons. Since the solitons are unstable, a natural question is which initial data evolve into each of the asymptotic forms. We consider the simplest setting in general space dimensions: the global behavior of solutions starting near a superposition of two ground states.

The main result is a complete classification of those solutions into 5 types of global behavior. Two of them are asymptotic to the positive ground state and the negative one respectively. They form two manifolds of codimension-1 that are joined at their boundary, which is the manifold of solutions asymptotic to superposition of two solitons. The connected union of those three manifolds separates the other solutions into the open set of global decaying solutions and that of blow-up. The manifold of 2-solitons was constructed by Cote, Martel, Yuan and Zhao. To get the classification, the main difficulty is in controlling the direction of instability attached to the two soliton components, because the soliton interactions are not integrable in time, breaking the simple linearized approximation. It is resolved by showing that the non-integrable interactions do not essentially affect the direction of instability, using the reflection symmetry of the equation and the 2-solitons.

I will also talk about a much harder difficulty in the 3-soliton case,
which may be called soliton merger.