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

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 [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.

Appendix Classical examples of local solutions to linear PDEs

Classically, mathematicians studied ordinary and partial differential equations from the point of view of calculating local solutions. That is, they were looking for formulas for analytic functions that solve the equations. Due to the advent of computers, it has become popular to think of ODEs and PDEs in terms of functions spaces of solutions. One of the most elementary classical solution is the local solution is d’Alembert’s solution to the (1+1)D wave equation.

The (1+1)D wave equation $u_{tt} = c u_{xx}$ has the local d’Alembert solution

$$\begin{aligned} u(x,t) = f(x-ct)+g(x+ct). \end{aligned}$$

(64)

In imaginary velocity case $c = i$, this equation becomes the elliptic 2D Laplace equation $u(x,y) = f(x+iy) + g(x-iy)$ where f(w) and g(w) solve the Cauchy–Riemann equations

$$\begin{aligned} \frac{\partial f}{\partial \overline{w}} = 0. \end{aligned}$$

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This approach generalizes to Whittaker’s solutions to the (2+1)D wave equation $u_{tt} = c^2 \Delta u$ and the 3D Laplace equation $\Delta u = 0$ is the case of imaginary time [13, 14]. Whittaker’s solution to the (2+1)D wave equation is

$$\begin{aligned} u(x,y,t) = \int _0^{2\pi } f(\cos (\theta )x + \sin (\theta )y + ct, \theta ) d \theta \end{aligned}$$

(66)

and Whittaker’s solution to the 3D Laplace equation is

$$\begin{aligned} u(x,y,z) = \int _0^{2\pi } f(\cos (\theta )x + \sin (\theta ) y + i z, \theta ) d \theta \end{aligned}$$

(67)

where $f(w,\theta )$ solves the Cauchy–Riemann equations

$$\begin{aligned} \frac{\partial f}{\partial \overline{w}} (w,\theta ) = 0. \end{aligned}$$

(68)

Expanding f as a Taylor series in w and a Fourier series in $\theta$ gives

$$\begin{aligned} f(z,\theta ) = \sum _{n,m=-\infty }^\infty a_{nm} w^m e^{i n \theta }, \end{aligned}$$

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and therefore, the general solution to the Laplace equation is the Whittaker series

$$\begin{aligned} u(x,y,z) = \sum _{n,m=-\infty }^\infty {a_{nm}} \int _{0}^{2\pi } \left( \cos (\theta ) x + \sin (\theta ) y + i z \right) ^m e^{in\theta } d \theta . \end{aligned}$$

(70)

The Whittaker series solution to the (2+1)D wave equation is

$$\begin{aligned} u(x,y,t) = \sum _{n,m=-\infty }^\infty {a_{nm}} \int _{0}^{2\pi } \left( \cos (\theta ) x + \sin (\theta ) y + ct \right) ^m e^{in \theta } d \theta . \end{aligned}$$

(71)

This series solution is a linear function of the coefficients $a_{nm}$. The Whittaker series allows the computation of solutions without boundary conditions.

The Whittaker series also generalize to the wave equation and Laplace equation in higher dimensions. The Whittaker series has the down side that the coefficients $a_{nm}$ do not uniquely determine the solution; however, it was used by Whittaker to justify the classical series solutions to the Laplace equation (series involving trigonometric functions, Bessel functions, spherical harmonics, etc.) before the advent of spectral theory. The usual spectral theory does not apply to nonlinear equations, so a local solution to the KP equation could help unify the inverse scattering transform for various boundary conditions.