DBAR problem

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.

References

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