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Stability of multisolitons for NLS


Academic Report

Title: Stability of multisolitons for NLS

Reporter: Prof. Herbert Koch (University of Bonn)

Time: Sept 26, 2019 (Thursday) PM 15:00-16:00

Location: A1101# room, Innovation Park Building

Contact: Prof. LI Junfeng (tel:84708351-1128)

Abstract: The focusing  nonlinear Schrödinger equation (NLS) has remarkable multisoliton solutions, related to eigenvalues of the corresponding Lax operator. Bäcklund transforms relate a neighborhood of the trivial solution to a neighborhood of solitons. Together with conserved energies this structure allows to prove stability of multisolitons. Novel features are
a) Uniform smoothness of the multisoliton set even in the case of spectral multiplicities, when the Bäcklund transform degenerates.
b) Stability of multisolitons even in the case of spectral multiplicity
c) Stability in fractional Sobolev spaces

The brief introduction to the reporter: Herbert Koch is a professor at the Institute of Mathematics, University of Bonn, Germany, and an internationally renowned expert in harmonic analysis and differential equations. The main research fields include real harmonic analysis, long-term behavior of dispersion equation solutions, and well-posedness theory of fluid equation solutions. His and Daniel Tataru's results on the well-posedness of solutions to Navier-Stockes equations are landmark results in this field, which are called Koch-Tataru solutions in the international mathematical community. The dual space theory of bounded variation space introduced by them has become an important workspace for modern harmonic analysis and the theory of nonlinear dispersion equation. Many non-linear problems provide new approaches; their recent studies on the conservation law of dispersion equation provide a new and powerful method for the study of long-term behavior of solutions of dispersion equation. Professor Herbert Koch has published more than 80 papers in top mathematical journals, including Duke Math, Adv. Math. At the same time, Professor Koch is also the editor of Analysis & PDE, and SIAM Journal of Mathematical Analysis.