- Source: DBAR problem
The DBAR problem, or the
∂
¯
{\displaystyle {\bar {\partial }}}
-problem, is the problem of solving the differential equation
∂
¯
f
(
z
,
z
¯
)
=
g
(
z
)
{\displaystyle {\bar {\partial }}f(z,{\bar {z}})=g(z)}
for the function
f
(
z
,
z
¯
)
{\displaystyle f(z,{\bar {z}})}
, where
g
(
z
)
{\displaystyle g(z)}
is assumed to be known and
z
=
x
+
i
y
{\displaystyle z=x+iy}
is a complex number in a domain
R
⊆
C
{\displaystyle R\subseteq \mathbb {C} }
. The operator
∂
¯
{\displaystyle {\bar {\partial }}}
is called the DBAR operator:
∂
¯
=
1
2
(
∂
∂
x
+
i
∂
∂
y
)
{\displaystyle {\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
∂
=
∂
∂
z
=
1
2
(
∂
∂
x
−
i
∂
∂
y
)
{\displaystyle \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
{\displaystyle z}
-plane.
The DBAR problem is of key importance in the theory of integrable systems, Schrödinger operators and generalizes the Riemann–Hilbert problem.
Citations
References
Ablowitz, Mark J.; Fokas, A. S. (2003). Complex Variables: Introduction and Applications. Cambridge University Press. pp. 516, 598–626. ISBN 978-0-521-53429-1.
Haslinger, Friedrich (2014). The d-bar Neumann Problem and Schrödinger Operators. Walter de Gruyter GmbH & Co KG. ISBN 978-3-11-031535-6.[1]
Konopelchenko, B. G. (2000). "On dbar-problem and integrable equations". arXiv:nlin/0002049.
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