Partially parity-time-symmetric ($\mathrm{p}\mathcal{PT}$-symmetric) lattice solitons are explored in quadratic nonlinear media. The solution of the model a nonlinear Schrödinger (NLS) equation with coupling to a mean term and an additional external potential, is computed by modern numerical methods, and it is shown that $\mathrm{p}\mathcal{PT}$-symmetric lattice solitons can exist in quadratic nonlinear media. The study concentrates on effects generated by the variation of lattice depth and quadratic nonlinearity strength that specify characteristics of the model, and the stability of the model is examined comprehensively by the nonlinear evolution and linear stability spectra of the solitons. It is demonstrated that stable evolution of solitons in a quadratic nonlinear media is possible for self-focusing $\mathrm{p}\mathcal{PT}$-symmetric lattices. Moreover, it is observed that, for the defocusing case of the lattice, fundamental solitons decay into radiation modes, and the decay of these solitons can be delayed by a deeper potential.

中文翻译：

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二次非线性介质中的部分PT对称格孤子

部分奇偶时间对称（$\mathrm{p}\mathcal{PT}$对称非线性晶格孤子是在二次非线性介质中探索的。通过现代数值方法计算了模型的解，该模型具有耦合到均值项和附加外部电势的非线性薛定ding（NLS）方程，并且证明：$\mathrm{p}\mathcal{PT}$对称非线性孤子可以存在于二次非线性介质中。研究集中在晶格深度和二次非线性强度变化所产生的影响上，这些变化确定了模型的特征，并通过孤子的非线性演化和线性稳定性谱对模型的稳定性进行了全面研究。证明了孤子在二次非线性介质中的稳定演化对于自聚焦是可能的$\mathrm{p}\mathcal{PT}$不对称晶格。此外，观察到，对于晶格的散焦情况，基本孤子衰减成辐射模式，并且这些孤子的衰减可以被更深的电位延迟。