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On solutions to the nonlocal \(\overline{\partial}\)-problem and (2+1) dimensional completely integrable systems. (English) Zbl 1462.32052

Lett. Math. Phys. 111, No. 1, Paper No. 16, 13 p. (2021); correction ibid. 111, No. 2, Paper No. 47, 2 p. (2021).
Summary: 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 [P. Nabelek, “Algebro-Geometric Finite Gap Solutions to the Korteweg-de Vries Equation as Primitive Solutions”, Phys. D 414, 132709 (2020); P. Nabelek and V. Zakharov, “Solutions to the Kaup-Broer system and its \((2+1)\) dimensional integrable generalization via the dressing method”, Phys. D 409, 132478 (2020)] 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.

MSC:

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
35Q53 KdV equations (Korteweg-de Vries equations)
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