Eduard Kirr : Computer Assisted Proofs for Large Bound States in Nonlinear Schrödinger Equations

Posted by Christof Sparber , part of the Analysis and Applied Mathematics Seminar.

At

Nov. 25, 2024, 4 p.m.

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Abstract

In an effort to find all bound states supported by the nonlinear Schrödinger equation we discovered that as their frequency approaches infinity each bound state separates into peaks which behave like particles. More precisely each peak localizes at a point in space and the force exerted by the potential depends solely on its position while the forces exerted by the other peaks depend on their relative positions. For the bound state to exist all peaks must be stationary hence the forces acting on it must sum to zero. Thus we get a system of algbraic equations which has unique solutions near a local minima of the potential but has infinitely many solutions near a saddle or local maxima. For the latter cases we employ a computer assisted proof to determine and classify a large set of solutions. This is joint work with A. Zarnescu (BCAM) and D. Manea (U. Bucharest).