Geodesic
Local algebraic analysis of differential systems
Teoretičeskaâ i matematičeskaâ fizika, Tome 183 (2015) no. 3, pp. 342-358 Cet article a éte moissonné depuis la source Math-Net.Ru

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We propose a new approach for studying the compatibility of partial differential equations. This approach is a synthesis of the Riquier method, Gröbner basis theory, and elements of algebraic geometry. As applications, we consider systems including the wave equation and the sine-Gordon equation.
Keywords: compatibility of differential equations, reduction, Gröbner basis. 
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O. V. Kaptsov. Local algebraic analysis of differential systems. Teoretičeskaâ i matematičeskaâ fizika, Tome 183 (2015) no. 3, pp. 342-358. http://geodesic.mathdoc.fr/item/TMF_2015_183_3_a0/

[1] J. Molk (ed.), Encyclopédie des sciences mathématiques. II, v. 4, Équations aux dérivées partielles, Jacques Gabay, Paris, 1916

[2] M. E. Goursat, Leçons sur l'intégration des équations aux dérivées partielles du second order à deux variables indépendantes, v. II, Librairie scientifique A. Hermann, Paris, 1898

[3] M. Kuranishi, Lectures on Involutive Systems of Partial Differential Equations, Publ. Soc. Mat., São Paulo, 1967 | Zbl

[4] D. Spenser, “Pereoperedelennye sistemy lineinykh differentsialnykh uravnenii v chastnykh proizvodnykh”, Sb. perevodov, Matematika, 14, vyp. 3, Mir, M., 1970, 99–126

[5] Zh.-F. Pommare, Sistemy uravnenii s chastnymi proizvodnymi i psevdogruppy Li, Mir, M., 1983 | MR | Zbl

[6] A. M. Vinogradov, “Geometriya nelineinykh differentsialnykh uravnenii”, Itogi nauki i tekhn. Ser. Probl. geom., 11, VINITI, M., 1980, 89–134 | MR | Zbl

[7] A. D. Wittkopf, Algorithms and implementations for differential elimination, PhD. Thesis, Simon Fraser University, Burnaby, British Columbia, Canada, 2004

[8] M. Marvan, Found. Comput. Math., 9:6 (2009), 651–674 | DOI | MR | Zbl

[9] V. P. Gerdt, Yu. A. Blinkov, “Janet-like monomial division”, Computer Algebra in Scientific Computing/CASC 2005 (Kalamata, Greece, September 12–16, 2005), Lecture Notes in Computer Science, 3718, Springer, Berlin, 2005, 174–183 | DOI | MR | Zbl

[10] B. Bukhberger, “Bazisy Grebnera. Algoritmicheskii metod v teorii polinomialnykh idealov”, Kompyuternaya algebra. Simvolnye i algebraicheskie vychisleniya, eds. B. Bukhberger, Dzh. Kollinz, R. Loos, Mir, M., 1986, 331–372 | MR

[11] D. Koks, Dzh. Littl, D. O'Shi, Idealy, mnogoobraziya i algoritmy, Mir, M., 2000 | DOI | MR | Zbl

[12] L. V. Ovsyannikov, Gruppovoi analiz differentsialnykh uravnenii, Nauka, M., 1978 | MR | Zbl

[13] N. Kh. Ibragimov, Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR | Zbl

[14] A. F. Sidorov, V. P. Shapeev, N. N. Yanenko, Metod differentsialnykh svyazei i ego prilozheniya v gazovoi dinamike, Nauka, Novosibirsk, 1984 | MR

[15] V. I. Fuschich, A. G. Nikitin, Simmetriya uravnenii kvantovoi mekhaniki, Nauka, M., 1990 | MR | Zbl

[16] M. Ablovits, Kh. Sigur, Solitony i metod obratnoi zadachi, Mir, M., 1987 | MR | MR | Zbl

[17] M. Atya, I. Makdonald, Vvedenie v kommutativnuyu algebru, Mir, M., 1972 | MR | Zbl

[18] G. Grauert, R. Remmert, Analiticheskie lokalnye algebry, Nauka, M., 1988 | MR | MR | Zbl

[19] J. Ritt, Differential Algebra, AMS, New York, 1950 | MR | Zbl

[20] O. V. Kaptsov, Programmirovanie, 40:2 (2014), 32–40 | DOI | MR

[21] O. Zarisskii, P. Samyuel, Kommutativnaya algebra, v. 2, IL, M., 1963 | MR | Zbl

[22] L. M. Brekhovskikh, Volny v sloistykh sredakh, Nauka, M., 1973 | MR | Zbl

[23] O. V. Kaptsov, Metody integrirovaniya uravnenii s chastnymi proizvodnymi, Fizmatlit, M., 2009

[24] V. B. Matveev, M. A. Salle, Darboux Trasformations and Solitons, Springer, New York, 1991 | MR | Zbl