In the Turnagain Arm in Alaska a famous tidal bore can be observed. Dispersive shockwaves can form in tidal bores leading to the formation of oscillations amplifying the bore. The Korteweg--de Vries (KdV) equation and the Kaup--Broer (KB) system are nonlinear PDEs describing long shallow water waves, and are completely integrable systems with an infinite number of degrees of freedom. Complete integrability allows these PDEs to be solved by two related methods based on: Solving a nonlocal d-bar problem or solving a Riemann--Hilbert problem. The focus of the talk will be on solutions that can be interpreted as nonlinear superpositions of an infinite number of soliton or cnoidal waves. Solutions produced by these methods include multi-soliton waves, periodic waves, and dispersive shockwaves.
The complete integrability of the KdV equation manifests itself in the fact that a solution to the KdV equation is a time dependent flow of one-dimensional potential energies such that the Schrödinger equations determined by all of the potential energies in the flow give the same energy spectrum. Therefore, the methods used to analyze KdV equation can be used to study the inverse problem of finding families of one dimensional potential energies such that the Schrödinger equation gives a prescribed energy spectrum.
This talk will present my own research, as well as research done in collaboration with Vladimir Zakharov, Kenneth McLaughlin, Dmitry Zakharov and Sergey Dyachenko.