The geometry of the space of Cauchy data of nonlinear PDEs. (English) Zbl 1292.35011

The author develops a framework for an intrinsic study of Cauchy problems for general nonlinear partial differential equations. The main point is a generalisation of the construction of the first-order jet bundle via Grassmannians using flag bundles and the subsequent introduction of higher- and infinite-order bundles. These new bundles allow for a simultaneous treatment of solutions of differential equations and their Cauchy data. As an application, transversality conditions in the calculus of variations are studied.


35A30 Geometric theory, characteristics, transformations in context of PDEs
14M15 Grassmannians, Schubert varieties, flag manifolds
35G25 Initial value problems for nonlinear higher-order PDEs
53B15 Other connections
58A20 Jets in global analysis
Full Text: DOI arXiv


[1] Bocharov A.V., Chetverikov V.N., Duzhin S.V., Khor’kova N.G., Krasil’shchik I.S., Samokhin A.V., Torkhov Yu.N., Verbovetsky A.M., Vinogradov A.M., Symmetries and Conservation Laws for Differential Equations of Mathematical Physics, Transl. Math. Monogr., 182, American Mathematical Society, Providence, 1999; · Zbl 0911.00032
[2] Bott R., Tu L.W., Differential Forms in Algebraic Topology, Grad. Texts in Math., 82, Springer, New York-Berlin, 1982 http://dx.doi.org/10.1007/978-1-4757-3951-0; · Zbl 0496.55001
[3] van Brunt B., The Calculus of Variations, Universitext, Springer, New York, 2004; · Zbl 1039.49001
[4] Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior Differential Systems, Math. Sci. Res. Inst. Publ., 18, Springer, New York, 1991 http://dx.doi.org/10.1007/978-1-4613-9714-4; · Zbl 0726.58002
[5] Giaquinta M., Hildebrandt S., Calculus of Variations. I, Grundlehren Math. Wiss., 310, Springer, Berlin, 1996; · Zbl 0853.49002
[6] Kijowski J., A simple derivation of canonical structure and quasi-local Hamiltonians in general relativity, Gen. Relativity Gravitation, 1997, 29(3), 307-343 http://dx.doi.org/10.1023/A:1010268818255; · Zbl 0873.53070
[7] Krasil’shchik J., Verbovetsky A., Geometry of jet spaces and integrable systems, J. Geom. Phys., 2011, 61(9), 1633-1674 http://dx.doi.org/10.1016/j.geomphys.2010.10.012; · Zbl 1230.58005
[8] Krupka D., Of the structure of the Euler mapping, Arch. Math. (Brno), 1974, 10(1), 55-61; · Zbl 0337.33012
[9] Michor P.W., Manifolds of Differentiable Mappings, Shiva Mathematics Series, 3, Shiva Publishing, Nantwich, 1980; · Zbl 0433.58001
[10] Moreno G., A C-spectral sequence associated with free boundary variational problems, In: Geometry, Integrability and Quantization, Avangard Prima, Sofia, 2010, 146-156; · Zbl 1382.58014
[11] Vinogradov A.M., Many-valued solutions, and a principle for the classification of nonlinear differential equations, Dokl. Akad. Nauk SSSR, 1973, 210, 11-14 (in Russian); · Zbl 0306.35003
[12] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory, J. Math. Anal. Appl., 1984, 100(1), 1-40 http://dx.doi.org/10.1016/0022-247X(84)90071-4; · Zbl 0548.58014
[13] Vinogradov A.M., The C-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory, J. Math. Anal. Appl., 1984, 100(1), 41-129 http://dx.doi.org/10.1016/0022-247X(84)90072-6; · Zbl 0548.58015
[14] Vinogradov A.M., Geometric singularities of solutions of nonlinear partial differential equations, In: Differential Geometry and its Applications, Brno, 1986, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987, 359-379; · Zbl 0631.58027
[15] Vinogradov A.M., Cohomological Analysis of Partial Differential Equations and Secondary Calculus, Transl. Math. Monogr., 204, American Mathematical Society, Providence, 2001; · Zbl 1152.58308
[16] Vinogradov A.M., Moreno G., Domains in infinite jet spaces: the C-spectral sequence, Dokl. Math., 2007, 75(2), 204-207 http://dx.doi.org/10.1134/S1064562407020081; · Zbl 1154.58002
[17] Vitagliano L., Secondary calculus and the covariant phase space, J. Geom. Phys., 2009, 59(4), 426-447 http://dx.doi.org/10.1016/j.geomphys.2008.12.001; · Zbl 1171.53057
[18] Vitagliano L., private communication, 2010;
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.