On the complete integrability of space-time shifted nonlocal equations

Abstract

We investigate the complete integrability of soliton equations with shifted nonlocal reductions under the rapidly decreasing boundary conditions. The illustrative examples we choose are the Ablowitz-Ladik (AL) system and the Ablowitz-Kaup-Newell-Segur (AKNS) system. For this two models with the space and space-time shifted nonlocal reductions, we establish the complete integrability of the resulting nonlocal systems by an explicit construction of the variables of action-angle type from the corresponding scattering data. Moreover, we find that the time shifted nonlocal reductions, unlike the space and space-time shifted ones, are not compatible with the Poisson bracket relations of the corresponding scattering data in the presence of the discrete spectrum.