Geodesic
Integrable structures for a generalized Monge–Ampère equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 2, pp. 208-224 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider a third-order generalized Monge–Ampère equation $u_{yyy}- u_{xxy}^2+u_{xxx}u_{xyy}=0$, which is closely related to the associativity equation in two-dimensional topological field theory. We describe all integrable structures related to it: Hamiltonian, symplectic, and also recursion operators. We construct infinite hierarchies of symmetries and conservation laws.
Mots-clés : Monge–Ampère equation, jet space
Keywords: integrability, Hamiltonian operator, symplectic structure, symmetry, conservation law, WDVV equation, two-dimensional topological field theory. 
@article{TMF_2012_171_2_a2,
     author = {A. M. Verbovetsky and R. Vitolo and P. Kersten and I. S. Krasil'shchik},
     title = {Integrable structures for a~generalized {Monge{\textendash}Amp\`ere} equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {208--224},
     year = {2012},
     volume = {171},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2012_171_2_a2/}
}
TY  - JOUR
AU  - A. M. Verbovetsky
AU  - R. Vitolo
AU  - P. Kersten
AU  - I. S. Krasil'shchik
TI  - Integrable structures for a generalized Monge–Ampère equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2012
SP  - 208
EP  - 224
VL  - 171
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2012_171_2_a2/
LA  - ru
ID  - TMF_2012_171_2_a2
ER  - 
%0 Journal Article
%A A. M. Verbovetsky
%A R. Vitolo
%A P. Kersten
%A I. S. Krasil'shchik
%T Integrable structures for a generalized Monge–Ampère equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2012
%P 208-224
%V 171
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2012_171_2_a2/
%G ru
%F TMF_2012_171_2_a2
A. M. Verbovetsky; R. Vitolo; P. Kersten; I. S. Krasil'shchik. Integrable structures for a generalized Monge–Ampère equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 171 (2012) no. 2, pp. 208-224. http://geodesic.mathdoc.fr/item/TMF_2012_171_2_a2/

[1] A. Kushner, V. Lychagin, V. Rubtsov, Contact Geometry and Nonlinear Differential Equations, Encyclopedia of Mathematics and Its Applications, 101, Cambridge Univ. Press, Cambridge, 2007 | MR | Zbl

[2] G. Boillat, C. R. Acad. Sci. Paris, Ser. Math., 315:11 (1992), 1211–1214 | MR | Zbl

[3] B. Dubrovin, “Geometry of 2-D topological field theories”, Integrable Systems and Quantum Groups (Montecatini Terme, Italy, June 14–22, 1993), Lecture Notes in Mathematics, 1620, Springer, Berlin, 1996, 120–348, arXiv: hep-th/9407018 | DOI | MR | Zbl

[4] E. V. Ferapontov, C. A. P. Galvão, O. I. Mokhov, Y. Nutku, Commun. Math. Phys., 186:3 (1997), 649–669 | DOI | MR | Zbl

[5] O. I. Mokhov, E. V. Ferapontov, Funkts. analiz i ego pril., 30:3 (1996), 62–72 ; O. I. Mokhov, E. V. Ferapontov, Equations of associativity in two-dimensional topological field theory as integrable hamiltonian nondiagonalizable systems of hydrodynamic type, arXiv: hep-th/9505180 | DOI | MR | Zbl

[6] J. Kalayci, Y. Nutku, Phys. Lett. A, 227:3–4 (1997), 177–182 | DOI | MR | Zbl

[7] J. Kalayci, Y. Nutku, J. Phys. A, 31:2 (1998), 723–734, arXiv: hep-th/9810076 | DOI | MR | Zbl

[8] I. A. B. Strachan, Phys. Lett. A, 210:4–5 (1996), 267–272 | DOI | MR | Zbl

[9] P. Kersten, I. Krasil'shchik, A. Verbovetsky, J. Geom. Phys., 50:1–4 (2004), 273–302, arXiv: math/0304245 | DOI | MR | Zbl

[10] I. Krasil'shchik, A. Verbovetsky, J. Geom. Phys., 61:9 (2011), 1633–1647, arXiv: 1002.0077 | DOI | MR

[11] V. A. Golovko, I. S. Krasil'shchik, A. M. Verbovetsky, Acta Appl. Math., 101:1–3 (2008), 59–83, arXiv: 0812.4681 | DOI | MR | Zbl

[12] P. Kersten, I. Krasil'shchik, A. Verbovetsky, Acta Appl. Math., 90:1–2 (2006), 143–178, arXiv: nlin/0511012 | DOI | MR | Zbl

[13] C. Crnkowić, E. Witten, “Covariant description of canonical formalism in geometrical theories”, Three Hundred Years of Gravitation, eds. S. W. Hawking, W. Israel, Cambridge Univ. Press, Cambridge, 1987, 676–684 | MR | Zbl

[14] G. J. Zuckerman, “Action principles and global geometry”, Mathematical Aspects of String Theory (San Diego, California, 1986), Advanced Series in Mathematical Physics, 1, ed. S. T. Yau, Word Scientific, Singapore, 1987, 259–284 | MR | Zbl

[15] A. M. Vinogradov, I. S. Krasilschik (red.), Simmetrii i zakony sokhraneniya uravnenii matematicheskoi fiziki, KhKh vek. Matematika i mekhanika, Faktorial Press, M., 2005 | MR

[16] A. M. Vinogradov, Dokl. AN SSSR, 236:2 (1977), 1200–1204 | Zbl

[17] A. M. Vinogradov, Dokl. AN SSSR, 238:5 (1978), 1028–1031 | MR | Zbl

[18] A. M. Vinogradov, J. Math. Anal. Appl., 100:1 (1984), 1–40 | DOI | MR | Zbl