The soliton solutions on both constant and periodic backgrounds of the nonlocal Davey–Stewartson III equation are derived by using the bilinear method and the Kadomtsev-Petviashvili (KP) hierarchy reduction method. These solutions are presented as [Formula: see text] Gram-type determinants, with [Formula: see text] a positive integer. Typical dynamics of these soliton solutions are investigated in analytical and graphical aspects. Two types of soliton solutions are generated with different [Formula: see text]. When [Formula: see text] is even, solitons on the constant background can be constructed, whereas solitons appear on the periodic background for odd [Formula: see text]. Under suitable parameter restrictions, we show the regularity of solutions and display all patterns of two- and four-soliton solutions.

中文翻译：

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非局域 Davey-Stewartson III 方程的孤子解

通过使用双线性方法和 Kadomtsev-Petviashvili (KP) 层次化约简方法，导出了非局部 Davey-Stewartson III 方程的恒定和周期性背景上的孤子解。这些解决方案表示为 [公式：见文本] 革兰式行列式，其中 [公式：见文本] 是一个正整数。这些孤子解的典型动力学在分析和图形方面进行了研究。用不同的[公式：见文本]生成两种类型的孤子解决方案。当[公式：见正文]为偶数时，可以构造常数背景上的孤子，而孤子出现在周期背景上的奇数[公式：见正文]。在合适的参数限制下，我们展示了解的规律性并展示了二孤子和四孤子解的所有模式。