Dr. Cheng Zhang
Shanghai University, China
On the inverse scattering transform for integrable PDEs on the half-line
In this talk, we provide an inverse scattering transform for integrable PDEs on the half-line. The method is based on the notions of integrable boundary conditions as well as the Sklyanin-type double-row monodromy matrix. Taking the nonlinear Schr”odinger (NLS) equation as our primary example. Following the double-row monodromy matrix formalism, integrable boundary conditions for NLS are encoded into constraints between the so-called reflection matrices and the time-part of the Lax pair of NLS. This gives rise to a hierarchy of reflection matrices, which is accompanied by a hierarchy of integrable boundary conditions. Then, by establishing a scattering system for the double-row monodromy matrix for NLS on the half-line, we obtain possible analytic and spectral properties of the scattering functions. This allows us to set up the inverse description of the half-line problem with integrable boundary conditions in terms of a Riemann-Hilbert problem. Solutions to the Riemann-Hilbert problems lead to solutions of NLS on the half-line subject to arbitrary-order integrable boundary conditions. In particular, multi-soliton solutions can be derived for the model.
Thu 6 May 2021 12PM Sydney time