Transactions of the American Mathematical Society

Author(s): Jeanne N. Clelland; Thomas A. Ivey
Journal: Trans. Amer. Math. Soc. 357 (2005), 1061-1093.
MSC (2000): Primary 37K35, 58J72; Secondary 35L10, 53C10, 58A15
Posted: July 16, 2004
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Abstract: We begin an exploration of parametric Bäcklund transformations for hyperbolic Monge-Ampère systems. (The appearance of an arbitrary parameter in the transformation is a feature of several well-known completely integrable PDEs.) We compute invariants for such transformations and explore the behavior of four examples, two of which are new, in terms of their invariants, symmetries, and conservation laws. We prove some preliminary results and indicate directions for further research.

1.: R. Bryant, S.-S. Chern, R. Gardner, P. Griffiths, H. Goldschmidt, Exterior Differential Systems, MSRI Publications, Springer, 1989.
2.: R. Bryant, P. Griffiths, Characteristic Cohomology of Differential Systems (II): Conservation Laws for a Class of Parabolic Equations, Duke Math. J. 78 (1995), 531-676. MR 96d:58158
3.: R. Bryant, P. Griffiths and L. Hsu, Hyperbolic exterior differential systems and their conservation laws, Part 1, Selecta Mathematica, New Series 1 (1995) 21-112. MR 97d:58008
4.: S.-S. Chern, C.-L. Terng, An analogue of Bäcklund's theorem in affine geometry, Rocky Mountain Math J. 10 (1980), 105-124. MR 81h:58004
5.: J.N. Clelland, A Bäcklund transformation for timelike surfaces of constant mean curvature in $\mathbb{R}^{1,2}$ , Bäcklund and Darboux Transformations. The Geometry of Solitons, 141-150, CRM Proc. Lecture Notes 29, Amer. Math. Soc., Providence, RI, 2001. MR 2002j:53011
6.: J. N. Clelland, Homogeneous Bäcklund transformations of hyperbolic Monge-Ampère systems, Asian J. Math. 6 (2002), 433-480. MR 2003j:58002
7.: P. Drazin, N. Johnson, Solitons: an introduction, Cambridge, 1989.
8.: E. Goursat, Leçons sur l'intégration des équations aux dérivées partielles du second ordre, vol. II, Gauthier-Villars, 1890.
9.: E. Goursat, Recherches sur quelques équations aux dérivées partielles du second ordre, Annales de la Faculté de Toulouse, deuxième serie 1 (1899), 31-78.
10.: S. Igonin, J. Krasil'shchik, On one-parametric families of Bäcklund transformations, Adv. Stud. Pure Math., vol. 37, Math. Soc. Japan, Tokyo, 2002.
11.: T.A. Ivey, J.M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, Graduate Studies in Mathematics, American Mathematical Society, 2003.
12.: B. O'Neill, Elementary Differential Geometry, 2nd edition, Academic Press, 1997.
13.: -, Semi-Riemannian Geometry, Academic Press, 1983. MR 85f:53002
14.: G. Penn-Karras, Klassification linearer Weingartenfläche in Raumformen, dissertation, Technische Universität Berlin, 1999.
15.: C. Rogers, Bäcklund transformations in soliton theory, in ``Soliton theory: a survey of results'', ed. A. P. Fordy, St. Martin's Press, 1990.
16.: C. Rogers, W. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, 1982. MR 84c:58002
17.: M.Y. Zvyagin, Second order equations reducible to $z_{xy}=0$ by a Bäcklund transformation, Soviet Math. Dokl. 43 (1991) 30-34. MR 93a:35010
18.: M. Wadati, H. Sanuki, and K. Konno, Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws. Progr. Theoret. Phys. 53 (1975), 419-436. MR 51:7516

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 37K35, 58J72, 35L10, 53C10, 58A15

Jeanne N. Clelland
Affiliation: Department of Mathematics, 395 UCB, University of Colorado, Boulder, Colorado 80309-0395
Email: Jeanne.Clelland@colorado.edu

Thomas A. Ivey
Affiliation: Department of Mathematics, College of Charleston, 66 George St., Charleston, South Carolina 29424-0001
Email: IveyT@cofc.edu

DOI: 10.1090/S0002-9947-04-03536-6
PII: S 0002-9947(04)03536-6
Keywords: B\"acklund transformations, hyperbolic Monge-Amp\`ere systems, Weingarten surfaces, exterior differential systems, Cartan's method of equivalence
Received by editor(s): May 8, 2003
Received by editor(s) in revised form: September 4, 2003
Posted: July 16, 2004
Copyright of article: Copyright 2004, American Mathematical Society

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