Seminar
The effect of positive bound states on the KdV solutions
by Alexei Rybkin (U. of Alaska Fairbanks)
Location: B203/5 (MdeHemptinne), Louvain-la-Neuve
Time: Tuesday June 4, 2019 at
15:00
We consider a slowly decaying oscillatory potential such that the corresponding 1D Schrödinger operator has a positive eigenvalue embedded into the absolutely continuous spectrum. This potential does not fall into a known class of initial data for which the Cauchy problem for the Korteweg-de Vries (KdV) equation can be solved by the inverse scattering transform. We nevertheless show that the KdV equation with our potential does admit a closed form classical solution in terms of Hankel operators. Comparing with rapidly decaying initial data our solution gains a new term responsible for the positive eigenvalue. To some extend this term resembles a positon (singular) solution but remains bounded. Our approach is based upon certain limiting arguments and techniques of Hankel operators.
