Riemann–Hilbert problems and inverse scattering of nonlocal real reverse-spacetime matrix AKNS hierarchies

Abstract

We would like to propose a kind of nonlocal real reverse-spacetime integrable hierarchies of PT-symmetric matrix AKNS equations through nonlocal symmetry reductions on the potential matrix, and formulate their associated Riemann–Hilbert problems to determine generalized Jost solutions of arbitrary-order matrix spectral problems. The Sokhotski–Plemelj formula is applied in transforming the associated Riemann–Hilbert problems into Gelfand–Levitan–Marchenko type integral equations. The Riemann–Hilbert problems corresponding to the reflectionless case are solved explicitly, where eigenvalues could equal adjoint eigenvalues, and thus, soliton solutions are presented for the resulting nonlocal real reverse-spacetime integrable PT-symmetric matrix AKNS equations.