Integrable space-time shifted nonlocal nonlinear equations

Abstract

In 1974 Ablowitz, Kaup, Newell, Segur (AKNS) put forward a theoretical framework whereby one can construct evolution equations that are (i) integrable in the sense of existence of infinite number of conservation laws and (ii) solvable by the inverse scattering transform. In subsequent years, many physically important integrable evolution equations were identified and the focus of the subject shifted towards methods to find special solutions and enhancing the underlying analysis. The discovery of a new reduction of the original AKNS system and the PT symmetric integrable nonlocal nonlinear Schrödinger (NLS) equation more than forty years later was surprising. Subsequently, additional nonlocal integrable reductions were found allowing nonlocality to be manifested in the time domain as well. This paper reports on yet another novel set of integrable reductions for the original AKNS system and associated new space-time nonlocal NLS type equations with space and time shifts. Integrability and inverse scattering transform are established along with soliton solutions. Their unique properties are discussed along with detailed comparison with the respective standard (non shifted) PT and reverse space-time symmetric NLS equations.