Research Article
The Cauchy Problem for Space-Time Monopole Equations in
Temporal and Spatial Gauge
Hyungjin Huh and Jihyun Yim
Department of Mathematics, Chung-Ang University, Seoul 156-756, Republic of Korea
Correspondence should be addressed to Hyungjin Huh; huh@cau.ac.kr
Received 7 October 2016; Revised 6 January 2017; Accepted 22 January 2017; Published 20 February 2017
Academic Editor: Ming Mei
Copyright © 2017 Hyungjin Huh and Jihyun Yim. is is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
We prove global existence of solution to space-time monopole equations in one space dimension under the spatial gauge condition
1
=0 and the temporal gauge condition
0
=0.
1. Introduction
In the current article, we study the following space-time
monopole equations in one space dimension:
+ [
0
, ] +
+ [
1
, ] = 0,
+ [
0
, ] +
+ [
1
, ] = 0,
1
−
0
+ [
0
,
1
] = [, ] .
(1)
Here , , = (
0
,
1
): R
1+1
→ g, where g is a Lie algebra
of a matrix Lie group such as SO(), SU() with Lie bracket
[⋅, ⋅]. We denote space-time derivatives by
0
=
,
1
=
.
e space-time monopole equations in R
2+1
can be
written as follows:
0
+
1
2
−
2
1
= [,
0
] + [
2
,
1
],
0
2
+
1
−
2
0
= [
2
,
0
] + [,
1
],
0
1
−
1
0
−
2
= [
2
, ] + [
1
,
0
].
(2)
Equation (1) is obtained by the dimensional reduction of
system (2). More precisely, we consider (2) independent of
the coordinate and renaming
2
as to get (1). e space-
time monopole system (2) is a nonabelian gauge field theory
and can be derived by dimensional reduction from anti-self-
dual Yang-Mills equations; see [1], for instance. e system
is an example of a completely integrable system and has an
equivalent formulation as a Lax pair. It was first introduced by
Ward in [2] as a hyperbolic analog of Bogomol’nyi equations
and discussed from the point of view of twistors.
System (1) is invariant under the rescaling
(, ) = (, ) ,
(, ) =
(, ) ,
(, ) = (, ) ,
(3)
from which we deduce a scale invariant Lebesgue space
1
(R)
and Sobolev space
̇
−1/2
(R). Another important property of
system (1) is an invariance under the gauge transformation
→
−1
,
→
−1
+
−1
,
→
−1
,
(4)
where : R
1+1
→ G is smooth and compactly supported
map into Lie group G.
A broad survey on the space-time monopole equations is
given in [1]. In particular, using the inverse scattering trans-
form, they have shown global existence and uniqueness up
to a gauge transformation for small initial data in
2,1
(R
2
).
e survey [1] also contained a number of other interesting
Hindawi
Advances in Mathematical Physics
Volume 2017, Article ID 4109645, 9 pages
https://doi.org/10.1155/2017/4109645