The DBAR problem, or the

\bar

-problem, is the problem of solving the

differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

\bar f (z, \bar) = g(z)

for the function

f(z,\bar)

, where

g(z)

is assumed to be known and

z = x + iy

is a complex number in a

domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...

R\subseteq \Complex

. The operator

\bar

is called the DBAR operator

\bar = \frac \left(\frac + i \frac \right)

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

\partial=\frac = \frac \left(\frac - i \frac \right)

denoting the usual differentiation in the complex

z

-plane. The DBAR problem is of key importance in the theory of

integrable systems In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...

and generalizes the Riemann–Hilbert problem.

References

{{Reflist

DBAR problem The DBAR problem, or the \bar-problem, is the problem of solving the differential equation \bar f (z, \bar) = g(z) for the function f(z,\bar), where g(z) is assumed to be known and z = x + iy is a complex number in a domain R\subseteq \Complex. The ...