Interplay between the inverse scattering method and the Fokas method and applications

The inverse scattering method (ISM), celebrating its 50th anniversary this year, was introduced in the seminal 1967 paper by Gardner, Greene, Kruskal, Miura as a spectral method to solve the Korteveg-de Vries equation, a nonlinear PDE modeling shallow water surface waves. It quickly developed into a systematic framework, nonlinearizing the classical Fourier transform to to solve certain nonlinear PDEs, now known as integrable PDEs. An important limitation of the ISM is the class of boundary conditions that one can impose one the problem. The search for a generalization of this method to include general initial-boundary value problems for integrable PDEs led to another breakthrough in the field: the so-called Unified Transform or Fokas method.

Dr Caudrelier will review both frameworks, their interplay and how they can be applied to current active areas: nonlocal reductions of integrable PDEs and nonlinear PDEs on networks. All the discussions will be illustrated with the so-called nonlinear Schrödinger equation.