Nonlocal continuous Hirota equation: Darboux transformation and symmetry broken and unbroken soliton solutions

The subject of this paper is a nonlocal Hirota equation. Firstly, we provide associated Lax pair and zero curvature condition to establish the integrability. Secondly, we construct N-fold Darboux transformation (DT) by taking the form of determinants. Thirdly, we derive parity-time (PT) symmetric broken bright soliton solutions under zero background and PT symmetric unbroken dark (or antidark) soliton solutions under plane wave background and simulate dynamic behaviors of those solutions. Respectively, we call solitons with instability as symmetry broken solitons and with stability as symmetry unbroken solitons. The root why two kinds of solitons occur is eigenvalue choices, leading to self-induced potential’s change. For bright solitons, potential terms both show unstable states, while interestingly their product (namely self-induced potential) is stable with the same parameter values. For dark and antidark solitons, potentials and their product all show stable states, and we present possible collision combinations of two potentials with the help of DT.