Abstract
We study the problem of the compatibility of nonlinear partial differential equations. We introduce the algebra of convergent power series, the module of derivations of this algebra, and the module of Pfaffian forms. Systems of differential equations are given by power series in the space of infinite jets. We develop a technique for studying the compatibility of differential systems analogous to the Gröbner bases. Using certain assumptions, we prove that compatible systems generate infinite manifolds.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Cox, J. Little, and D. O’Shea, Ideals, Varieties, and Algorithms: An introduction to Computational Algebraic Geometry and Commutative Algebra, Springer, New York (1997).
B. Buchberger, “Gröbner bases: An algorithmic method in polynomial ideal theory,” in: Progress, Directions, and Open Problems in Multidimensional Systems Theory (N. K. Bose, ed.), D. Reidel, Dordrecht (1985), pp. 184–232.
C. Riquier, Les systèmes d’équations aux dérivées partielles, Gauthier-Villars, Paris (1909).
M. Janet, Leçons sur les systémes des équations aux dérivées partielles, Gauthier-Villars, Paris (1929).
J. F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups, Gordon and Breach, New York (1978).
A. M. Vinogradov, I. S. Krasil’schik, and V. V. Lychagin, Introduction to the Geometry of Nonlinear Differential Equations [in Russian], Nauka, Moscow (1986).
V. V. Zharinov, Lecture Notes on Geometrical Aspects of Partial Differential Equations (Series Sov. East Eur. Math., Vol. 9), World Scientific, Singapore (1992).
W. M. Seiler, Involution: The Formal Theory of Differential Equations and Its Applications in Computer Algebra (Algor. Comput. Math., Vol. 24), Springer, New York (2010).
M. Marvan, Found. Comput. Math., 9, 651–674 (2009).
Maple, http://www.maplesoft.com/products/Maple/index.aspx (2015).
O. V. Kaptsov, Program. Comput. Softw., 40, 63–70 (2014).
O. V. Kaptsov, Theor. Math. Phys., 183, 740–755 (2015).
H. Grauert and R. Remmert, Analytische Stellenalgebren (Grundlehren Math. Wiss., Vol. 176), Springer, Berlin (1971).
R. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, New York (1965).
J.-P. Serre, Lie Algebras and Lie Groups (Lect. Notes Math., Vol. 1500), Springer, Berlin (1992).
N. Bourbaki, Algèbre, Hermann, Paris (1962).
L. V. Ovsyannikov, Group Analysis of Differential Equations [in Russian], Nauka, Moscow (1978).
A. M. Vinogradov and I. S. Krasil’schik, eds., Symmetries and Conservation Laws of Equations of Mathematical Physics [in Russian], Faktorial, Moscow (1997).
V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachov, and A. A. Rodionov, Applications of Group-Theoretical Methods in Hydrodynamics, Springer, Amsterdam (2010).
N. Bourbaki, General Topology, Springer, Berlin (1998).
A. I. Belousov and S. B. Tkachev, Discrete Mathematics [in Russian], Bauman Moscow State Technical Univ., Moscow (2004).
J. F. Ritt, Differential Algebra, Dover, New York (1966).
D. Mumford, The Red Book of Varieties and Schemes (Lect. Notes Math., Vol. 1358), Springer, Berlin (1999).
C.-S. Yih, Stratified Flows, Acad. Press, New York (1980).
Yu. V. Shan’ko, Vychisl. Tekhnol., 6, No. 5, 106–117 (2001).
M. Golubitsky and V. Guillemin, StableMappings and Their Singularities (Grad. TextsMath., Vol. 14), Springer, New York (1973).
V. A. Dorodnitsyn, J. Soviet Math., 55, 1490–1517 (1991).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 189, No. 2, pp. 219–238, November, 2016.
This research was performed under the financial support of a grant from the Russian government for the conduct of research under the direction of leading scientists at the Siberian Federal University (Contract No. 14.U26.31.006) and the Program for Supporting Leading Scientific Schools (Grant Nos. NSh-544.2012.1 and NSh-6293.2012.9).
Rights and permissions
About this article
Cite this article
Kaptsov, O.V. Algebraic and geometric structures of analytic partial differential equations. Theor Math Phys 189, 1592–1608 (2016). https://doi.org/10.1134/S0040577916110052
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577916110052