On a class of perturbations of the special pole-free joint solution of KdV and PI2 under the action of the KdV flow

by Alexander Minakov (UCLouvain)

Location: CYCL09b, Louvain-la-Neuve
Time: Thursday October 18, 2018 at 15:00

We consider perturbations of the special pole-free joint solution \(U(x,t)\) of the Korteweg-de Vries equation \(u_t+uu_x+\frac{1}{12}u_{xxx}=0\) and \(P_I^2\) equation \(u_{xxxx}+10u_x^2+20uu_{xx}+40(u^3-6tu+6x)=0\) under the action of the KdV flow. We show that if the perturbation is compact and of bounded variation, then the initial value problem for the KdV equation has a classical solution. Our method is the inverse scattering transform method in the form of the Riemann-Hilbert problem method. Namely, we construct the corresponding spectral functions \(a(\lambda), r(\lambda),\) and give characterization of the compact perturbations in terms of \(a(\lambda), r(\lambda)\). This is a joint work in progress with B. Dubrovin.