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.

本文的主题是一个非局部 Hirota 方程。首先，我们提供相关的 Lax 对和零曲率条件来建立可积性。其次，我们构造N-fold Darboux 变换 (DT) 通过采用行列式的形式。第三，我们推导出零背景下的奇偶时间（PT）对称破碎亮孤子解和平面波背景下的 PT 对称完整暗（或反暗）孤子解，并模拟这些解的动态行为。分别，我们将具有不稳定性的孤子称为对称破缺的孤子，将稳定的称为对称未破的孤子。两种孤子出现的根本原因是特征值的选择，导致自诱导势的变化。对于明亮的孤子，势项都表现出不稳定的状态，而有趣的是，它们的乘积（即自感应势）在相同的参数值下是稳定的。对于暗孤子和反暗孤子，势及其乘积都显示稳定状态，