| Title: Integrable Initial Boundary Value Problems Revisited
The Inverse Scattering Transform (IST) method is a powerful tool for studying initial value (IV) problems for integrable PDEs.
The adaptation of the method to initial boundary value (IBV) problems
for integrable equations goes back to the late 80s, when special classes of
boundary conditions were singled out (works of Bikbaev, Fokas, Sklyanin, Tarasov
and the speaker). These boundary conditions preserve the complete integrality
of a system. They allow an appropriate, based on the Backlund transformation,
continuation of the given initial data on, say, a half line to the whole axis and hence a
reduction of the IBV problem to an appropriate IV problem.
An alternative formalism to the IBV problems for interable PDEs was suggested
in the early 90s by A. Fokas. Fokas s method is based on the Riemann-Hilbert
approach and, in principle, can be applied to arbitrary boundary conditions.
The integrable boundary conditions play a special role in Fokas s method as well.
Indeed, it is only for these boundary conditions the method yields a complete linearization
of the problem.
In the talk, which is based on the joint work with D. Shepelsky, a somewhat novel
point of view on the integrable IBV problems will be presented. Our approach is a
combination of both Fokas s method and some of the ideas developed in the 80s.
We will consider the IBV for the nonlinear Shr dingier equation on the half-line with
Robin boundary condition as a case study. A brief review of some other recent works on
the subject will be also attempted, if time permits.
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