Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-14T04:41:23.423Z Has data issue: false hasContentIssue false
  • English
  • Français

GROUP FOLIATION OF DIFFERENTIAL EQUATIONS USING MOVING FRAMES

Part of:Pseudogroups, differentiable groupoids and general structures on manifoldsRepresentations of solutionsClassical differential geometryPartial differential equations

Published online by Cambridge University Press:  26 October 2015

ROBERT THOMPSON and
FRANCIS VALIQUETTE
ROBERT THOMPSON
Affiliation:
Department of Mathematics, Carleton College, Northfield, MN 55057, USA; rthompson@carleton.edu
FRANCIS VALIQUETTE
Affiliation:
Department of Mathematics, SUNY at New Paltz, New Paltz, NY 12561, USA

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We incorporate the new theory of equivariant moving frames for Lie pseudogroups into Vessiot’s method of group foliation of differential equations. The automorphic system is replaced by a set of reconstruction equations on the pseudogroup jets. The result is a completely algorithmic and symbolic procedure for finding both invariant and noninvariant solutions of differential equations admitting a symmetry group.

MSC classification

Primary:35B06: Symmetries, invariants, etc.53A55: Differential invariants (local theory), geometric objects
Secondary:35C05: Solutions in closed form58H05: Pseudogroups and differentiable groupoids

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 3 , 2015, e22
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

References

Anderson, I. M., ‘The variational bicomplex’, Technical Report, Utah State University, 2000.Google Scholar
Anderson, I. M. and Fels, M. E., ‘Transformations of Darboux integrable systems’, inDifferential Equations: Geometry, Symmetries and Integrability, Abel Symp., 5 (Springer, Berlin, 2009), 21–48.CrossRefGoogle Scholar
Anderson, I. A. and Fels, M. E., ‘Exterior differential systems with symmetry’, Acta Appl. Math. 87 (2005), 3–31.Google Scholar
Bobenko, A. I. and Suris, Y. B., Discrete Differential Geometry: Integrable Structure, Graduate studies in Mathematics, 98 (American Mathematical Society, Providence, RI, 2008).Google Scholar
Calogero, F., ‘A solvable nonlinear wave equation’, Stud. Appl. Math. 70(3) (1984), 189–200.Google Scholar
Cartan, É., ‘Les groupes de transformations continus, infinis, simples’, Ann. Éc. Norm. 26 (1909), 93–161.Google Scholar
Champagne, B. and Winternitz, P., ‘On the infinite-dimensional symmetry group of the Davey–Stewartson equations’, J. Math. Phys. 29(1) (1988), 1–8.Google Scholar
Chan, T. F. and Shen, J., ‘Non-texture inpainting by curvature-driven diffusions’, J. Visual Comm. Image Rep. 12 (2001), 436–449.Google Scholar
Chirkunov, Y. A. and Prudnikov, V. Y., ‘Group reduction of the Lamé equations’, Prikl. Mat. Mekh. 52(3) (1988), 366–371.Google Scholar
Coley, A. A., (ed.) Bäcklund and Darboux Transformations: The Geometry of Solitons: AARMS–CRM Workshop, June 4–9, 1999, Halifax, NS, Canada, Vol. 29, American Mathematical Society, 2001.Google Scholar
David, D., Levi, D. and Winternitz, P., ‘Bäcklund transformations and the infinite-dimensional symmetry group of the Kadomtsev–Petviashvili equation’, Phys. Lett. A 118(8) (1986), 390–394.Google Scholar
Faucher, M. and Winternitz, P., ‘Symmetry analysis of the Infeld–Rowlands equation’, Phys. Rev. E 48 (1993), 3066–3071.Google Scholar
Fels, M. and Olver, P. J., ‘Moving coframes. II. Regularization and theoretical foundations’, Acta Appl. Math. 55 (1999), 127–208.Google Scholar
Guggenheimer, H. G., Differential Geometry, (McGraw-Hill, New York, 1963).Google Scholar
Hubert, E., ‘Differential invariants of a Lie group action: syzygies on a generating set’, J. Symbolic Comput. 44(4) (2009), 382–416.Google Scholar
Igonin, S., ‘Coverings and fundamental algebras for partial differential equations’, J. Geom. Phys. 56(6) (2006), 939–998.CrossRefGoogle Scholar
Itskov, V., Olver, P. J. and Valiquette, F., ‘Lie completion of pseudo-groups’, Transform. Groups 16 (2011), 161–173.CrossRefGoogle Scholar
Jacobson, N., Lie Algebras, (Dover, New York, 1979).Google Scholar
Johnson, H. H., ‘Classical differential invariants and applications to Partial differential equations’, Math. Ann. 148 (1962), 308–329.Google Scholar
Krasil’shchik, I. S. and Vinogradov, A. M., ‘Nonlocal symmetries and the theory of covering’, Acta Appl. Math. 2 (1984), 79–86.Google Scholar
Kruglikov, B. and Lychagin, V., ‘Invariants of pseudo-group actions: homological methods and finiteness theorem’, Int. J. Geom. Methods Mod. Phys. 3 (2006), 1131–1165.Google Scholar
Kruglikov, B. and Lychagin, V., ‘Global Lie–Tresse theorem’, 2015, arXiv:1111.5480v3.Google Scholar
Kumpera, A., ‘Invariants différentiels d’un pseudo-groupe de Lie’, J. Differ. Geom. 10 (1975), 289–416.Google Scholar
Langer, J. and Singer, D. A., ‘Lagrangian aspects of the Kirchhoff elastic rod’, SIAM Rev. 38(4) (1996), 605–618.CrossRefGoogle Scholar
Lei, H.-C., ‘Group splitting and linearization mapping of a solvable non-linear wave equation’, Int. J. Nonlinear Mech. 33(3) (1998), 461–471.Google Scholar
Lie, S. and Scheffers, G., Vorlesungen über Continuierliche Gruppen mit Geometrischen und Anderen Anwendungen, (B. G. Teubner, Leipzig, 1893).Google Scholar
Lie, S., ‘Zur allgemeinen Theorie der partiellen Differentialgleichungen beliebiger Ordnung’, Leipz. Ber. 1 (1895), 53–128.Google Scholar
Mansfield, E. L., A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics, (Cambridge University Press, Cambridge, 2010).Google Scholar
Mansfield, E. L. and van der Kamp, P. H., ‘Evolution of curvature invariants and lifting integrability’, J. Geom. Phys. 56(8) (2006), 1294–1325.Google Scholar
Martina, L., Sheftel, M. B. and Winternitz, P., ‘Group foliation and non-invariant solutions of the heavenly equation’, J. Phys. A 34 (2001), 9243–9263.Google Scholar
Medolaghi, P., ‘Classificazione delle equazioni alle derivative parziali del secondo ordine, che ammettono un gruppo infinito di trasformazioni puntuali’, Ann. Mat. Pura Appl. 3 1898001 (1898), 229–263.Google Scholar
Munoz, J., Muriel, F. J. and Rodriguez, J., ‘On the finiteness of differential invariants’, J. Math. Anal. Appl. 284 (2003), 266–282.Google Scholar
Niethammer, M., Tannenbaum, A. and Angenent, S., ‘Dynamic active contours for visual tracking’, IEEE Trans. Automat. Control 51 (2006), 562–579.CrossRefGoogle ScholarPubMed
Nutku, Y. and Sheftel, M. B., ‘Differential invariants and group foliation for the complex Monge–Ampère equation’, J. Phys. A 34 (2001), 137–156.Google Scholar
Olver, P. J., Applications of Lie Groups to Differential Equations, 2nd edn, (Springer, New York, 1993).Google Scholar
Olver, P. J., Equivalence, Invariants, and Symmetry, (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Olver, P. J., ‘Moving frames and joint differential invariants’, Regul. Chaotic Dyn. 4(4) (1999), 3–18.CrossRefGoogle Scholar
Olver, P. J., ‘Joint invariant signatures’, Found. Comput. Math. 1 (2001), 3–67.Google Scholar
Olver, P. J., ‘Generating differential invariants’, J. Math. Anal. Appl. 333 (2007), 450–471.Google Scholar
Olver, P. J., ‘Lectures on moving frames’, inSymmetries and Integrability of Difference Equations, (eds. Levi, D., Olver, P., Thomova, Z. and Winternitz, P.) London Mathematical Society Lecture Note Series, 381 (Cambridge University Press, Cambridge, 2011), 207–246.Google Scholar
Olver, P. J. and Pohjanpelto, J., ‘Maurer–Cartan forms and the structure of Lie pseudo-groups’, Selecta Math. 11 (2005), 99–126.Google Scholar
Olver, P. J. and Pohjanpelto, J., ‘Differential invariants for Lie pseudo-groups’, Comput. Appl. Math. 1 (2007), 1–27.Google Scholar
Olver, P. J. and Pohjanpelto, J., ‘Moving frames for Lie pseudo-groups’, Can. J. Math. 60 (2008), 1336–1386.Google Scholar
Olver, P. J. and Pohjanpelto, J., ‘Differential invariant algebras of Lie pseudo-groups’, Adv. Math. 222 (2009), 1746–1792.Google Scholar
Ondich, J., ‘The reducibility of partially invariant solutions of systems of partial differential equations’, Eur. J. Math. 6(4) (1995), 329–354.Google Scholar
Olver, P. J. and Valiquette, F., ‘Recursive moving frames for Lie pseudo-groups’, Preprint SUNY New Paltz, 2014.Google Scholar
Ovsiannikov, L. V., Group Analysis of Differential Equations, (Academic Press, New York, 1982).Google Scholar
Pohjanpelto, J., ‘Reduction of exterior differential systems with infinite dimensional symmetry groups’, BIT Num. Math. 48 (2008), 337–355.Google Scholar
Rodrigues, A. A. M., ‘On infinite Lie groups’, Ann. Inst. Fourier 31(3) (1981), 245–274.CrossRefGoogle Scholar
Sapiro, G., Geometric Partial Differential Equations and Image Analysis, (Cambridge University Press, Cambridge, 2001).Google Scholar
Stormark, O., Lie’s Structural Approach to PDE Systems, Encyclopedia of Mathematics and its Applications, 80 (Cambridge University Press, New York, 2000).Google Scholar
Thompson, R., ‘Applications of moving frames to group foliation of differential equations’, PhD Thesis, University of Minnesota, 2013.Google Scholar
Thompson, R. and Valiquette, F., ‘Group foliation of finite difference equations’ (in preparation).Google Scholar
Tresse, A., ‘Sur les invariants différentiels des groupes continus de transformations’, Acta Math. 18 (1894), 1–88.Google Scholar
Valiquette, F., ‘Solving local equivalence problems with the equivariant moving frame method’, SIGMA 9 (2013), 029.Google Scholar
Vessiot, E., ‘Sur l’intégration des systèmes différentiels qui admettent des groupes continus de transformations’, Acta Math. 28 (1904), 307–349.Google Scholar
Yau, S.-T., ‘Calabi’s conjecture and some new results in algebraic geometry’, Proc. Natl Acad. Sci. USA 74(5) (1977), 1798–1799.Google Scholar