On the complete integrability of space–time shifted nonlocal equations

We investigate the complete integrability of soliton equations with shifted nonlocal reductions under rapidly decaying boundary conditions. Using the Ablowitz–Ladik (AL) system and the Ablowitz–Kaup–Newell–Segur (AKNS) system as illustrative examples, we establish the complete integrability of models with space and space–time shifted nonlocal reductions through the explicit construction of canonical action–angle variables from their scattering data. Moreover, we demonstrate that, unlike the space and space–time shifted nonlocal cases, time-shifted nonlocal reductions are incompatible with the Poisson bracket structures of the scattering data in the presence of discrete spectrum.