Nonlocal PT-symmetric integrable equations and related Riemann–Hilbert problems

We aim to discuss about how to construct and classify nonlocal PT-symmetric integrable equations via nonlocal group reductions of matrix spectral problems. The nonlocalities considered are reverse-space, reverse-time and reverse-spacetime, each of which can involve either the transpose or the Hermitian transpose. The associated spectral problems are used to formulate a kind of Riemann–Hilbert problems and thus inverse scattering transforms. Soliton solutions are generated from specific Riemann–Hilbert problems with the identity jump matrix. We focus on two expository examples: nonlocal PT-symmetric matrix nonlinear Schrödinger and modified Korteweg–de Vries equations.