On the complete integrability of space-time shifted nonlocal equations

TL;DR

This paper demonstrates the complete integrability of space and space-time shifted nonlocal soliton equations, specifically the AL and AKNS systems, by constructing action-angle variables from scattering data.

Contribution

It establishes the integrability of shifted nonlocal reductions of AL and AKNS systems through explicit construction of action-angle variables.

Findings

Space and space-time shifted nonlocal reductions are integrable.

Time shifted nonlocal reductions are incompatible with Poisson brackets.

Explicit action-angle variables are constructed from scattering data.

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.