DBAR problem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The DBAR problem is the problem of solving the differential equation

\bar{\partial} f (z,\bar{z}) = g(z)

for the function f(z,\bar{z}), where g(z) is assumed to be known and z=x+iy is a complex number in a domain R\subseteq \mathbb{C}. The operator \bar{\partial} is called the DBAR operator

\bar{\partial} = \frac{1}{2} \left(\frac{\partial}{\partial x} + i \frac{\partial}{\partial y} \right)

The DBAR operator is nothing other than the complex conjugate of the operator

\partial=\frac{\partial}{\partial z}=\frac{1}{2} \left(\frac{\partial}{\partial x} - i \frac{\partial}{\partial y} \right)

denoting the usual differentiation in the complex z-plane.

The DBAR problem is of key importance in the theory of integrable systems[1] and generalizes the Riemann–Hilbert problem.


  1. ^ Konopelchenko, B. G. "On dbar-problem and integrable equations". arXiv:nlin/0002049.