Integrable nonlocal Hirota equations

TL;DR

This paper introduces new integrable nonlocal versions of the Hirota equation, constructs their explicit solutions, and explores their unique properties and types of soliton solutions.

Contribution

It develops novel nonlocal integrable models related to the Hirota equation and provides explicit multi-soliton solutions using Hirota's method and Darboux transformations.

Findings

Constructed new nonlocal Hirota-type equations with proven integrability.

Derived explicit multi-soliton solutions exhibiting nonlocal and time crystal behaviors.

Identified diverse solution types, including rogue waves and nonlocal in space/time structures.

Abstract

We construct several new integrable systems corresponding to nonlocal versions of the Hirota equation, which is a particular example of higher order nonlinear Schr\"{o}dinger 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 $\mathcal{P}$, by a time reversal $\mathcal{T}$ or a $\mathcal{PT}$-transformation possibly combined with a conjugation. We construct explicit multi-soliton 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 behaviour. The multi-soliton 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 rogues wave type.