Graduate Seminar Analysis

 

Tuesday, July 18, 2017, 10:30 a.m

Conserved energies for the one-dimensional cubic nonlinear Schrödinger equations

Xian Liao, University of Bonn

Abstract: In this talk I will present the derivation of the conserved energies which are equivalent to the H^s-norms of the solutions of the one-dimensional cubic nonlinear Schrödinger equations (NLS), that is, all the H^s-norms (with some lower bound for s) of the solutions are conserved a priori: This is done in the recent exciting work by Koch-Tataru. I will also present briefly my recent progress with Professor Koch for the Gross-Pitaevskii equation (GP): (GP) is defocusing (NLS) but with a non standard constraint on the solution q at infinity: |q| -> 1 at infinity. technically this talk will be divided into three parts. In the introduction part, I will start with a rough statement about the conserved energies for (NLS) and (GP), and then introduce the essential notation: the transmission coefficient, and the three technical tools: the Hopf algebra, the frequency-rescaled norms and the superharmonic functions on the upper half plane. Then I will explain how to formulate the conserved energies by use of the transmission coefficient and how to show its equivalence to the H^s-norms by use of the technical tools: This is done by Koch-Tataru for (NLS). Finally for the Gross-Pitaevskii equation (GP) we will see that the non standard constraint at infinity causes essential new difficulties in the analysis.

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Ort:  Raum 008/SeMath, Pontdriesch 14-16, 52062 Aachen