On Solutions to the Nonlocal $\overline{\partial}$ Problem and (2+1) Dimensional Completely Integrable Systems

TL;DR

This paper introduces a new formula for solving the nonlocal bar-problem, applies it to complex (2+1)D integrable systems, and discusses solutions including solitons and classical wave equations, advancing methods in integrable systems theory.

Contribution

It presents a novel formula for the nonlocal bar-problem and demonstrates its application to complex (2+1)D integrable equations, connecting dressing methods with classical solutions.

Findings

Explicit solutions to (2+1)D KP and Kaup--Broer systems.

Application of the formalism to finite gap and soliton solutions.

Analogy between dressing method and Whittaker solutions.

Abstract

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 [10], [8] 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 (2+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.