In this short note, we discuss a new formula for solving the nonlocal \(\overline{\partial }\)-problem, and discuss application to the Manakov–Zakharov dressing method. We then explicitly apply this formula to solving the complex (2+1)D Kadomtsev–Petviashvili equation and complex (2+1)D completely integrable generalization of the (2+1)D Kaup–Broer (or Kaup–Boussinesq) system. We will also discuss how real (1+1)D solutions are expressed using this formalism. It is simple to express the formalism for finite gap primitive solutions from [9, 11] using the formalism of this note. We also discuss recent results on the infinite soliton limit for the (1+1)D Korteweg–de Vries equation and the (1+1)D Kaup-Broer system. In an appendix, the classical solutions to the 3D Laplace equation (2+1)D d’Alembert wave equation by Whittaker are described. This appendix is included to elucidate an analogy between the dressing method and the Whittaker solutions.

在此简短说明中，我们讨论了用于解决非局部\（\ overline {\ partial} \）的新公式问题，并讨论在Manakov–Zakharov敷料方法中的应用。然后，我们明确地将此公式用于求解复数（2 + 1）D Kadomtsev-Petviashvili方程和复数（2 + 1）D对（2 + 1）D Kaup-Broer（或Kaup-Boussinesq）系统的完全可积分泛化。我们还将讨论使用这种形式主义如何表达真实的（1 + 1）D解决方案。使用本说明的形式主义，很容易从[9，11]中表达有限间隙原始解的形式主义。我们还将讨论关于（1 + 1）D Korteweg-de Vries方程和（1 + 1）D Kaup-Broer系统的无限孤子极限的最新结果。在附录中，描述了Whittaker对3D Laplace方程（2 + 1）D d'Alembert波方程的经典解。包括本附录是为了阐明修整方法和Whittaker解决方案之间的类比。