Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 27
where E 2 (t) is the Eisenstein series, and σ 1 (n) are particular number-theoretic
coefficients. From (5.15) or (5.16) the general solution is obtained from (5.13)
where a, b, c, d are taken to be arbitrary coefficients with ad − bc =1.Ifweuse
(5.12), then a convenient “Bäcklund” type transformation is
y II (t) =
y I(γt)
(ct + d) 2 − 6c
ct + d . (5.17)
Equations (5.12)-(5.17) demonstrate that the solutions to Chazy’s equation (5.11)
and the solution to the Darboux-Halphen system (5.10) (by finding w 1 ,w 2 ,w 3
in terms of y, ẏ, ÿ) are expressible in terms of automorphic functions.
An interesting question to ask is whether the solution of (5.10–5.11) can be
obtained via the inverse method. In fact, in a recent paper [23] it has been shown
that the linear compatible system of SDYM can be reduced to a monodromy
problem. The novelty is that in this case the monodromy problem has evolving
monodromy data+– unlike those associated with the Painlevé equation where
the monodromy is fixed (isomonodromy). Then the linear problem can be used
to find the solutions of (5.10) which are automorphic functions, and via (5.10), to
solve Chazy’s equation (5.11). Generalizations of the Darboux-Halphen system
are also examined and solved in [23]. It is outside the scope of this article to go
into those details.
Acknowledgments
This work was partially supported by the Air Force Office of Scientific Research,
Air Force Materials Command, USAF under grant F49620-97-1-0017, and by the
NSF under grant DMS-9404265. The US Government is authorized to reproduce
and distribute reprints for governmental purposes notwithstanding any copyright
notation thereon. The views and conclusions contained herein are those of the
author and should not be interpreted as necessarily representing the official
policies or endorsements, either expressed or implied, of the Air Force Office
of Scientific Research or the US Government.
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