Inverse scattering transform for continuous and discrete space-time shifted integrable equations

Nonlocal integrable partial differential equations possessing a spatial or temporal reflection have constituted an active research area for the past decade. Recently, more general classes of these nonlocal equations have been proposed, wherein the nonlocality appears as a combination of a shift (by a real or a complex parameter) and a reflection. This new shifting parameter manifests itself in the inverse scattering transform (IST) as an additional phase factor in an analogous way to the classical Fourier transform. In this paper, the IST is analyzed in detail for several examples of such systems. Particularly, time, space, and space-time shifted nonlinear Schr\"odinger (NLS) and space-time shifted modified Korteweg-de Vries (mKdV) equations are studied. Additionally, the semi-discrete IST is developed for the time, space and space-time shifted variants of the Ablowitz-Ladik integrable discretization of the NLS. One soliton solutions are constructed for all continuous and discrete cases.