Geometry of Bäcklund transformations I: generality
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- by Yuhao Hu PDF
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Abstract:
Using Élie Cartan’s method of equivalence, we prove an upper bound for the generality of generic rank-1 Bäcklund transformations relating two hyperbolic Monge-Ampère systems. In cases when the Bäcklund transformation admits a symmetry group whose orbits have codimension 1, 2, or 3, we obtain classification results and new examples of auto-Bäcklund transformations.References
- I. M. Anderson and M. E. Fels, Symmetry reduction of exterior differential systems and Bäcklund transformations for PDE in the plane, Acta Appl. Math. 120 (2012), 29–60. MR 2945627, DOI 10.1007/s10440-012-9716-0
- I. M. Anderson and M. E. Fels, Bäcklund transformations for Darboux integrable differential systems, Selecta Math. (N.S.) 21 (2015), no. 2, 379–448. MR 3338681, DOI 10.1007/s00029-014-0159-5
- Albert V. Bäcklund, Om ytor med konstant negativ krokning, Lunds Universitets Arsskrift, 19, 1883.
- R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths, Exterior differential systems, Mathematical Sciences Research Institute Publications, vol. 18, Springer-Verlag, New York, 1991. MR 1083148, DOI 10.1007/978-1-4613-9714-4
- Robert Bryant, Phillip Griffiths, and Daniel Grossman, Exterior differential systems and Euler-Lagrange partial differential equations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2003. MR 1985469
- R. Bryant, P. Griffiths, and L. Hsu, Hyperbolic exterior differential systems and their conservation laws. I, Selecta Math. (N.S.) 1 (1995), no. 1, 21–112. MR 1327228, DOI 10.1007/BF01614073
- Robert L. Bryant, An introduction to Lie groups and symplectic geometry, Geometry and quantum field theory (Park City, UT, 1991) IAS/Park City Math. Ser., vol. 1, Amer. Math. Soc., Providence, RI, 1995, pp. 5–181. MR 1338391, DOI 10.1090/pcms/001/02
- Robert L. Bryant, Notes on exterior differential system, arXiv:1405.3116, 2014.
- Jeanne N. Clelland and Thomas A. Ivey, Parametric Bäcklund transformations. I. Phenomenology, Trans. Amer. Math. Soc. 357 (2005), no. 3, 1061–1093. MR 2110433, DOI 10.1090/S0002-9947-04-03536-6
- Jeanne N. Clelland and Thomas A. Ivey, Bäcklund transformations and Darboux integrability for nonlinear wave equations, Asian J. Math. 13 (2009), no. 1, 15–64. MR 2500957, DOI 10.4310/AJM.2009.v13.n1.a3
- Jeanne Nielsen Clelland, Homogeneous Bäcklund transformations of hyperbolic Monge-Ampère systems, Asian J. Math. 6 (2002), no. 3, 433–480. MR 1946343, DOI 10.4310/AJM.2002.v6.n3.a4
- Shiing Shen Chern and Chuu Lian Terng, An analogue of Bäcklund’s theorem in affine geometry, Rocky Mountain J. Math. 10 (1980), no. 1, 105–124. MR 573866, DOI 10.1216/RMJ-1980-10-1-105
- Robert B. Gardner, The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1062197, DOI 10.1137/1.9781611970135
- Édouard Goursat, Le probleme de Bäcklund (memorial des sciences mathematiques. fasc. vi), Paris: Gauthier-Villars, 1925.
- Yuhao Hu, Geometry of Backlund Transformations, ProQuest LLC, Ann Arbor, MI, 2018. Thesis (Ph.D.)–Duke University. MR 3818790
- J. J. C. Nimmo and D. G. Crighton, Bäcklund transformations for nonlinear parabolic equations: the general results, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 381–401. MR 684316, DOI 10.1098/rspa.1982.0164
- C. Rogers and W. K. Schief, Bäcklund and Darboux transformations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. Geometry and modern applications in soliton theory. MR 1908706, DOI 10.1017/CBO9780511606359
- Chuu-Lian Terng and Karen Uhlenbeck, Geometry of solitons, Notices Amer. Math. Soc. 47 (2000), no. 1, 17–25. MR 1733063
- Georges Tzitzéica, Sur une nouvelle classe de surfaces, Rendiconti del Circolo Matematico di Palermo (1884-1940), 25(1):180–187, 1908.
- Georges Tzitzéica, Sur une nouvelle classe de surfaces (2ème partie), Rendiconti del Circolo Matematico di Palermo (1884-1940), 28(1):210–216, 1909.
- Hugo D. Wahlquist and Frank B. Estabrook, Bäcklund transformation for solutions of the Korteweg-de Vries equation, Phys. Rev. Lett. 31 (1973), 1386–1390. MR 415100, DOI 10.1103/PhysRevLett.31.1386
Additional Information
- Yuhao Hu
- Affiliation: Department of Mathematics, 395 UCB, University of Colorado, Boulder, Colorado 80309-0395
- Email: Yuhao.Hu@colorado.edu
- Received by editor(s): February 26, 2019
- Received by editor(s) in revised form: June 25, 2019
- Published electronically: November 5, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 1181-1210
- MSC (2010): Primary 37K35, 35L10, 58A15, 53C10
- DOI: https://doi.org/10.1090/tran/7992
- MathSciNet review: 4068261