Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Parametric Bäcklund transformations I: Phenomenology

Authors: Jeanne N. Clelland and Thomas A. Ivey
Journal: Trans. Amer. Math. Soc. 357 (2005), 1061-1093
MSC (2000): Primary 37K35, 58J72; Secondary 35L10, 53C10, 58A15
Published electronically: July 16, 2004
MathSciNet review: 2110433
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 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

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 37K35, 58J72, 35L10, 53C10, 58A15

Additional Information

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

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

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
Published electronically: July 16, 2004
Article copyright: © Copyright 2004 American Mathematical Society