Rybalko Y. Inverse scattering transform method for nonlocal integrable equations

This thesis is devoted to the analysis of the initial value problems for integrable nonlocal nonlinear Schrodinger equation. We consider problems with two types of boundary conditions as spatial variable tends to infinity: (i) zero boundary conditions (decaying problems) and (ii) asymmetric step-like boundary conditions (problems with step-like initial data). For these problems we develop inverse scattering transform method in the form of the matrix Riemann-Hilbert problem, which has some new interesting and distinctive features, particularly, pseudo-residue condition on the contour of the problem and new type of “singular” points where certain spectral functions have the growing winding of the argument. For obtaining the long-time asymptotics we adapt the nonlinear steepest decent method to the corresponding basic Riemann-Hilbert problem.