Nonlocal nonlinear Schr\"odinger equation and its discrete version: soliton solutions and gauge equivalence

In this paper, we try to understand the geometry for a nonlocal nonlinear Schr\"{o}dinger equation (nonlocal NLS) and its discrete version introduced by Ablowitz and Musslimani. We show that, under the gauge transformations, the nonlocal focusing NLS and the nonlocal defocusing NLS are, respectively, gauge equivalent to a Heisenberg-like equation and a modified Heisenberg-like equation, and their discrete versions are, respectively, gauge equivalent to a discrete Heisenberg-like equation and a discrete modified Heisenberg-like equation. From the gauge equivalence, although the geometry related to the nonlocal NLS is not very clear, we can see that the properties between the nonlocal NLS and its discrete version and NLS and discrete NLS have big difference. By constructing the Darboux transformation for discrete nonlocal NLS equations including the cases of focusing and defocusing, we derive their discrete soliton solutions, which differ from the ones obtained by using the scattering transformation.