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arXiv · 2008.13237

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

Patrik V. Nabelek
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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.

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Patrik V. Nabelek. 2020-08-30. On Solutions to the Nonlocal $\overline{\partial}$ Problem and (2+1) Dimensional Completely Integrable Systems. https://doi.org/10.1007/s11005-021-01392-3

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