Integrable nonlocal Hirota equations

We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schrödinger equations. The integrability of the new models is established by providing their explicit forms of Lax pairs or zero curvature conditions. The two compatibility equations arising in this construction are found to be related to each other either by a parity transformation P, by a time reversal T or a P T-transformation possibly combined with a conjugation. We construct explicit multisoliton solutions for these models by employing Hirota’s direct method as well as Darboux-Crum transformations. The nonlocal nature of these models allows for a modification of these solution procedures as the new systems also possess new types of solutions with different parameter dependence and different qualitative behavior. The multisoliton solutions are of varied type, being, for instance, nonlocal in space, nonlocal in time of time-crystal type, regular with local structures either in time/space or of rogue wave type.

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