In this paper, the existence and stability of solitons in parity-time ($\mathcal{PT}$)-symmetric optical media characterized by a generic complex hyperbolic refractive index distribution with fourth-order diffraction (FOD) coefficient and higher-order nonlinearities have been investigated. For the linear case, we have demonstrated numerically that, the FOD parameter can alter the $\mathcal{PT}$-breaking points. Exact analytical expressions of the localized modes are obtained respectively, in one and two dimensional nonlinear Schrödinger (NLS) equation with both self-focusing and self-defocusing Kerr, and higher-order nonlinearities for nonlinear case. The effects of both FOD and higher-order nonlinearities on the stability/instability structure of these localized modes have also been discussed with the help of linear stability analysis followed by the direct numerical simulation of the governing equation. Some stable and unstable solutions have been given and, it has been seen how higher-order self-focusing and self-defocusing nonlinearities can influence the stability/instability of the system.

本文研究了奇偶时间孤子的存在性和稳定性（$\mathcal{PT}$)-对称光学介质的特点是具有四阶衍射 (FOD) 系数和高阶非线性的通用复双曲线折射率分布。对于线性情况，我们已经通过数值证明了 FOD 参数可以改变$\mathcal{PT}$- 突破点。分别在具有自聚焦和自散焦克尔的一维和二维非线性薛定谔（NLS）方程以及非线性情况下的高阶非线性中获得了局部模式的精确解析表达式。在线性稳定性分析和控制方程的直接数值模拟的帮助下，还讨论了 FOD 和高阶非线性对这些局部模式的稳定性/不稳定性结构的影响。已经给出了一些稳定和不稳定的解决方案，并且已经看到高阶自聚焦和自散焦非线性如何影响系统的稳定性/不稳定性。