Integrable nonlocal Hirota equations

Abstract

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 $\mathcal{P}$, by a time reversal $\mathcal{T}$ or a $\mathcal{P}\mathcal{T}$-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.

23 pages, 7 figures

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• Asymptotic analysis of high order soliton for the Hirota equation
• Asymptotic and scattering behaviour for degenerate multi-solitons in the Hirota equation
• Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems
• An introduction to PT-symmetric quantum mechanics -- time-dependent systems
• Rational solutions of the defocusing nonlocal nonlinear Schrodinger equation: Asymptotic analysis and soliton interactions
• Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background
• Novel Superposed Kinklike and Pulselike Solutions for Several Nonlocal Nonlinear Equations
• Nonlocal gauge equivalence: Hirota versus extended continuous Heisenberg and Landau-Lifschitz equation
• Stability in integrable nonlocal nonlinear equations
• Multicomplex solitons
• Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model
• Superposed periodic kink and pulse solutions of coupled nonlinear equations
• Conformal bridge transformation, $\mathcal{P}\mathcal{T}$- and super- symmetry
• Non-Hermitian superintegrable systems
• Darboux transformations and solutions of nonlocal Hirota and Maxwell-Bloch equations