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\begin{document}
\title{B\"acklund Transformations and Darboux Integrability for Nonlinear Wave Equations}
\author{Jeanne N. Clelland}
\address{Dept. of Mathematics, 395 UCB, University of
Colorado\\
Boulder, CO 80309-0395}
\email{Jeanne.Clelland@colorado.edu}
\author{Thomas A. Ivey}
\address{Dept. of Mathematics, College of Charleston\\
66 George St., Charleston SC 29424-0001}
\email{IveyT@cofc.edu}

\subjclass[2000]{Primary 37K35 58J72; Secondary 35L10, 37K35, 53C10, 58A15}
\keywords{B\"acklund transformations, hyperbolic Monge-Amp\`ere systems,
Weingarten surfaces, exterior differential systems, Cartan's method of equivalence}


\begin{abstract}
We prove that second-order \MA/ equations for one function of two variables
are connected to the wave equation by a \BT/ if and only if they are integrable
by the method of Darboux at second order.
\end{abstract}
\maketitle
%\baselineskip=18pt %for double spacing


\begin{comment}
1. Introduction [TAI]
   -background: results of Goursat, Zvyagin; state Darboux-Goursat theorem
(D-int at some level iff BT to wave at some level)
   -state overall program (or leave this to end of paper)
   -goal for paper is THM  BT to z_xy=0 iff D-int after one prolongation

2.  Setup [JNC]
   -G-structure, interpretation of invariants
   -getting M/A invariants in terms of G-structure invariants (only as much as we need, though)
   -holonomic, and why it's not interesting

3. BT implies Darboux [JNC]
   -break into cases 3a, 3b, 3c depending on the dimension of the second derived systems
of the characteristic systems on B^6

4. Darboux implies BT [???]
   -break into cases 4a, 4b depending on the size of the second derived system of characteristic systems on M^5
(case 4b may have to be split into quasilinear, semiquasilinear and nonlinear)

5. Examples [TAI]
   -solve for BT for s=p*exp(u) (and possibly also Liouville's equation)

6. Next Steps [TAI]
   -parametric (example: Zvyagin 3 /Goursat II)
   -other stuff?
\end{comment}


\input{wintro}

\input{wsetup}

\input{wprovedarboux}

\input{wprovebacklund}

\input{wapp}

\input{wconclusion}

\begin{thebibliography}{9}
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Ph.D. thesis, Utah State University, 2003.

\bibitem{BGG} R. Bryant, P. Griffiths, D. Grossmann,
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\bibitem{C1} J. Clelland, {\bf Homogeneous \BB/ transformations of
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\bibitem{CI} J. Clelland, T. Ivey, {\bf Parametric \BT/s I: Phenomenology},
{\em Trans. Amer. Math. Soc.} 357 (2005), 1061--1093.

\bibitem{G89} R. Gardner, {\bf The Method of Equivalence and Its Applications},
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\bibitem{GLecons} E. Goursat, {\bf Le\c{c}ons sur l'int\'egration des
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\bibitem{G1899} --, {\bf Recherches sur quelques \'equations aux
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{\em Annales de la Facult\'e de Toulouse (deuxi\`eme serie)} 1 (1899), 31--78.

\bibitem{CFB} T. Ivey, J.M. Landsberg, {\bf Cartan for Beginners: Differential geometry via moving frames and
exterior differential systems} (with J.M. Landsberg), Graduate Studies
in Mathematics vol. 61, American Mathematical Society, 2003.

\bibitem{Fordy} A.P. Fordy, {\em A Historical Introduction to
Solitons and B\"acklund Transformations}, in ``Harmonic Maps and Integrable
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\bibitem{Juras} M. Jur\'a\v{s}, {\bf Geometric Aspects of Second-Order Partial Differential Equations in the Plane},
Ph.D. thesis, Utah State University, 1997.

\bibitem{V1942} E. Vessiot, {\bf Sur les \'equations aux d\'eriv\'ees partielle du second ordre int\'egables par la methode de Darboux},
{\em J. Math Pures Appl.} 18 (1939), 1--61 and 21 (1942) 1--66.

\bibitem{Wu}
Xifang Cao, Hongyou Wu, Chuanyou Xu,
{\bf On Miura transformations among nonlinear partial differential equations},  J. Math. Phys.  47  (2006), no. 8.

\bibitem{Zvyagin1} M.Yu. Zvyagin, {\bf Classification of \BT/ of Second-Order Partial Differential Equations},
{\em Mat. Zametki}  29  (1981), 829--842 (English translation: {\em Math. Notes} 29 (1981), 422--429).

\bibitem{Zvyagin2} --, {\bf Second order equations reducible to
$z_{xy}=0$ by a \BB/ transformation}, {\em Dokl. Akad. Nauk SSSR} 316 (1991), 36--40 (English translation: {\em Soviet Math. Dokl.}
43 (1991) 30--34).

\end{thebibliography}

\end{document}