In this paper, we study the differential-difference Kadomtsev–Petviashvili (D$\Delta$KP) equation through the dressing method based on a nonlocal $\bar{\partial }$-problem. Cauchy type determinant solution and a new type rational solution are given. The initial value problem of D$\Delta$KP is discussed by virtue of the quasi-local $\bar{\partial }$-problem. From an impulse initial condition, a new explicit solution to the initial value problem of D$\Delta$KP is obtained.

中文翻译：

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差分KP方程的新解

在本文中，我们研究了Kadomtsev–Petviashvili（D$\Delta$KP）方程通过基于非局部的修整方法 $\bar{\partial }$-问题。给出了柯西型行列式解和一种新型的有理解。D的初值问题$\Delta$通过准局部讨论KP $\bar{\partial }$-问题。从脉冲初始条件，到D初始值问题的新显式解$\Delta$获得KP。