A continuous analog of the binary Darboux transformation for the Korteweg-de Vries equation

In the KdV context we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann-Hilbert problem and yields a new explicit formula for perturbation of the negative spectrum of a wide class of step-type potentials without changing the rest of the scattering data. This extends the previously known formulas for inserting/removing finitely many bound states to arbitrary sets of negative spectrum of arbitrary nature. In the KdV context our method offers same benefits as the classical binary Darboux transformation does.Comment: To appear in Studies in Applied Mat

Inverse problem for the nonlinear long wave runup on a plane sloping beach

We put forward a simple but effective explicit method of recovering initial data for the nonlinear shallow water system from the reading at the shoreline. We then apply our method to the tsunami waves inverse problem