Integrable Nonlocal Reductions

Abstract: We present some nonlocal integrable systems by using the Ablowitz-Musslimani nonlocal reductions. We first present all possible nonlocal reductions of nonlinear Schr\"{o}dinger (NLS) and modified Korteweg-de Vries (mKdV) systems. We give soliton solutions of these nonlocal equations by using the Hirota method. We extend the nonlocal NLS equation to nonlocal Fordy-Kulish equations by utilizing the nonlocal reduction to the Fordy-Kulish system on symmetric spaces. We also consider the super AKNS system and then show that Ablowitz-Musslimani nonlocal reduction can be extended to super integrable equations. We obtain new nonlocal equations namely nonlocal super NLS and nonlocal super mKdV equations.