In this paper, the Hirota's bilinear method and Kadomtsev‐Petviashvili hierarchy reduction method are applied to construct soliton, line breather and (semi‐)rational solutions to the nonlocal Mel'nikov equation with nonzero boundary conditions. These solutions are expressed as Gram‐type determinants. When N is even, soliton, line breather and (semi‐)rational solutions on the constant background are derived while these solutions are located on the periodic background for odd N. Regularity of these solutions and their connections with the local Mel'nikov equation are analyzed for proper choices of parameters that appear in the solutions. The dynamics of the solutions are discussed in detail. All possible configurations of soliton and lump solutions are found for . Several interesting dynamical behaviors of semi‐rational solutions are observed. It is shown that certain lumps may exhibit fusion and fission phenomena during their interactions with solitons while some lump may change its direction of movement after it collides with solitons.

中文翻译：

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周期背景下非局部梅尔尼科夫方程的一般孤子和（半）解

本文采用Hirota的双线性方法和Kadomtsev-Petviashvili层次化约简方法来构造边界条件为非零的非局部Mel'nikov方程的孤子，线性呼吸和（半）有理解。这些解决方案表示为Gram型行列式。当N为偶数时，在恒定背景下导出孤子，线呼吸和（半）有理解，而这些解位于奇数N的周期性背景下。分析这些解的规则性及其与局部Mel'nikov方程的联系，以选择出现在解中的参数。详细讨论了解决方案的动态。找到了孤子和块解决方案的所有可能配置。观察到了一些有趣的半理性解的动力学行为。结果表明，某些团块在与孤子相互作用时可能表现出融合和裂变现象，而某些团块在与孤子碰撞后可能改变其运动方向。