Methods of geometry of differential equations in analysis of integrable models of field theory

A. V. Kiselev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 1, pp. 57-165.

Résumé

In this paper, we investigate algebraic and geometric properties of hyperbolic Toda equations $u_{xy}=\exp(Ku)$ associated with nondegenerate symmetrizable matrices $K$. A hierarchy of analogues of the potential modified Korteweg"– de Vries equation $u_t=u_{xxx}+u_x^3$ is constructed and its relationship with the hierarchy for the Korteweg– de Vries equation $T_t=T_{xxx}+TT_x$ is established. Group-theoretic structures for the dispersionless $(2+1)$-dimensional Toda equation $u_{xy}=\exp(-u_{zz})$ are obtained. Geometric properties of the multi-component nonlinear Schrödinger equation type systems $\Psi_t=\boldsymbol i\Psi_{xx}+\boldsymbol if(|\Psi|)\Psi$ (multi-soliton complexes) are described.

Bibliographie

[1] Andreev V. A., “Preobrazovaniya Beklunda tsepochek Tody”, Teor. i matem. fiz., 75:3 (1988), 340–352 | MR

[2] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1979 | MR

[3] Arnold V. I., Geometricheskie metody v teorii obyknovennykh differentsialnykh uravnenii, Izhevskaya resp. tipogr., Izhevsk, 2000

[4] Bogolyubov N. N., Shirkov D. V., Kvantovye polya, Fizmatlit, M., 1993 | MR

[5] Bocharov A. V., Verbovetskii A. M., Vinogradov A. M. i dr., Simmetrii i zakony sokhraneniya uravnenii matematicheskoi fiziki, eds. A. M. Vinogradov i I. S. Krasilschik (red.), Faktorial, M., 1997

[6] Gelfand I. M., Dorfman I. Ya., “Gamiltonovy operatory i svyazannye s nimi algebraicheskie struktury”, Funktsion. analiz i ego pril., 13:4 (1979), 13–30 | MR

[7] Golovko V. A., “O zakonakh sokhraneniya dlya sistem Tody”, X Mezhdunarodnaya konferentsii studentov i aspirantov po fundamentalnym naukam “Lomonosov-2003”, sektsiya “Fizika”, Sbornik tezisov, MGU, M., 2003, 53–55

[8] Golovko V. A., “O predstavleniyakh nulevoi krivizny i preobrazovaniyakh Beklunda dlya uravneniya Liuvillya”, Trudy XXV Konferentsii molodykh uchenykh. Mekhaniko-matematicheskii f-t MGU, MGU, M., 2003, 20–22

[9] Demskoi D. K., Startsev S. Ya., “O postroenii simmetrii po integralam sistem giperbolicheskikh uravnenii”, Fundam. i prikl. mat., 10:1 (2004), 29–37 | MR | Zbl

[10] Drinfeld V. G., Sokolov V. V., “Uravneniya tipa Kortevega–de Friza i prostye algebry Li”, DAN SSSR, 258:1 (1981), 11–16 | MR

[11] Drinfeld V. G., Sokolov V. V., “Algebry Li i uravneniya tipa Kortevega–de Friza”, Itogi nauki i tekhn. Ser. Sovr. probl. matematiki. Noveishie dostizheniya, 24, VINITI, M., 1984, 81–180 | MR

[12] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya: Metody i prilozheniya, Nauka, M., 1979 | MR

[13] Zhiber A. V., “Uravneniya $n$-voln i sistema nelineinykh uravnenii Shrëdingera s gruppovoi tochki zreniya”, Teor. i matem. fiz., 52:3 (1982), 405–413 | MR | Zbl

[14] Zhiber A. V., Sokolov V. V., “Tochno integriruemye giperbolicheskie uravneniya liuvillevskogo tipa”, Uspekhi mat. nauk, 56:1 (2001), 63–106 | MR | Zbl

[15] Zhiber A. V., Shabat A. B., “Uravneniya Kleina–Gordona s netrivialnoi gruppoi”, DAN SSSR, 247:5 (1979), 1103–1107 | MR

[16] Zograf P. G., Takhtadzhyan L. A., “Ob uravnenii Liuvillya, aktsessornykh parametrakh i geometrii prostranstva Teikhmyullera dlya rimanovykh poverkhnostei roda 0”, Mat. sb., 132(174):2 (1987), 147–166 | MR | Zbl

[17] Ibragimov N. Kh., Shabat A. B., “Uravnenie Kortevega–de Friza s gruppovoi tochki zreniya”, DAN SSSR, 244:1 (1979), 57–61 | MR | Zbl

[18] Ibragimov N. Kh., Shabat A. B., “Evolyutsionnye uravneniya s netrivialnoi gruppoi Li–Beklunda”, Funktsion. analiz i ego pril., 14:1 (1980), 25–36 | MR | Zbl

[19] Kiselev A. V., “Klassicheskie zakony sokhraneniya dlya ellipticheskogo uravneniya Liuvillya”, Vestnik Mosk. un-ta. Ser. 3, Fizika, astronomiya, 2000, no. 6, 11–13 | MR

[20] Kiselëv A. V., “Ob avtopreobrazovanii Beklunda dlya uravneniya Liuvillya”, Vestnik Mosk. un-ta. Ser. 3, Fizika, astronomiya, 2002, no. 6, 22–26 | MR

[21] Kiselëv A. V., “O nekotorykh svoistvakh operatora rekursii dlya uravneniya Liuvillya”, Trudy XXV Konferentsii molodykh uchenykh. Mekhaniko-matematicheskii f-t MGU, Sb. tezisov, MGU, M., 2003, 74–77

[22] Kiselëv A. V., Metody geometrii differentsialnykh uravnenii v analize integriruemykh modelei teorii polya, Diss. $\dots$ kand. fiz. -mat. nauk, MGU, M., 2004

[23] Kiselëv A. V., “O zakonakh sokhraneniya v solitonnykh kompleksakh”, XXVI Konferentsiya molodykh uchenykh. Mekhaniko-matematicheskii fakultet MGU, Sb. tezisov, MGU, M., 2004, 62–63

[24] Kiselëv A. V., “O nepreryvnom analoge dvumernykh sistem Tody”, Mat. i eë pril., 1:1 (2004), 69–74

[25] Kiselëv A. V., “O nëterovykh simmetriyakh uravnenii Tody”, Vestnik Mosk. un-ta. Ser. 3, Fizika, astronomiya, 2004, no. 2, 16–18 | MR

[26] Kiselëv A. V., “O postroenii tochnykh reshenii bezdispersionnogo uravneniya Tody”, Mat. i eë pril., 1:2 (2004)

[27] Kiselev A. V., Ob uravneniyakh Kortevega–de Friza, assotsiirovannykh s sistemami Tody, Dep. v VINITI 10.03.2004, No412-B2004

[28] Kiselëv A. V., “Primenenie metodov geometrii differentsialnykh uravnenii v reshenii kraevykh zadach”, Mat. i eë pril., 1:1 (2004), 59–68

[29] Kiselëv A. V., “Ob assotsiativnykh algebrakh Shlezingera–Stashefa i opredelitelyakh Vronskogo”, Fundam. i prikl. mat., 11:1 (2005), 159–180 | MR | Zbl

[30] Kiselëv A. V., Ovchinnikov A. V., “O nekotorykh gamiltonovykh ierarkhiyakh, assotsiirovannykh s uravneniyami Tody”, Lomonosovskie chteniya-2004. Sektsiya fiziki, 2004, 102–105, MGU, M.

[31] Leznov A. N., “O polnoi integriruemosti odnoi nelineinoi sistemy differentsialnykh uravnenii v chastnykh proizvodnykh v dvumernom prostranstve”, Teor. i matem. fiz., 42:3 (1980), 343–349 | MR | Zbl

[32] Leznov A. N., Savelev M. V., Gruppovye metody integrirovaniya nelineinykh dinamicheskikh sistem, M., 1985 | MR

[33] Leznov A. N., Smirnov V. G., Shabat A. B., “Gruppa vnutrennikh simmetrii i usloviya integriruemosti dvumernykh dinamicheskikh sistem”, Teor. i matem. fiz., 51:1 (1982), 10–21 | MR | Zbl

[34] Meshkov A. G., “Simmetrii skalyarnykh polei. III: Dvumernye integriruemye modeli”, Teor. i matem. fiz., 63:3 (1985), 323–332 | MR | Zbl

[35] Dzhet Nestruev, Gladkie mnogoobraziya i nablyudaemye, MTsNMO, M., 2000

[36] Ovchinnikov A. V., Sistemy Tody, assotsiirovannye s algebrami Li, i $W$-algebra v nekotorykh zadachakh matematicheskoi fiziki, Diss. $\dots$ kand. fiz. -mat. nauk, MGU, M., 1996

[37] Polyakov A. M., Kalibrovochnye polya i struny, Izd-vo “Udmurtskii universitet”, Izhevsk, 1999 | Zbl

[38] Savelev M. V., “O probleme integriruemosti nepreryvnoi sistemy Tody”, Teor. i matem. fiz., 92:3 (1992), 457–465 | MR

[39] Toda M., Teoriya nelineinykh reshëtok, M., 1984 | MR

[40] Khorkova N. G., “Zakony sokhraneniya i nelokalnye simmetrii”, Mat. zametki, 44:1 (1988), 134–144 | MR

[41] Shabat A. B., Yamilov R. I., Eksponentsialnye sistemy tipa I i matritsy Kartana, Preprint, Bashkir. filial AN SSSR, Ufa, 1981

[42] Akhmediev N., Ankiewicz A., “Multi-soliton complexes”, Chaos, 10:3 (2000), 600–612 | DOI | MR | Zbl

[43] Alfinito E., Soliani G., Solombrino L., “The symmetry structure of the heavenly equation”, Lett. Math. Phys., 41 (1997), 379–389 | DOI | MR | Zbl

[44] Barnich G., Brandt F., Henneaux M., “Local BRST cohomology in the antifield formalism: I. General theorems”, Comm. Math. Phys., 174 (1995), 57–92 | DOI | MR

[45] Bilal A., Gervais J.-L., “Extended $C=\infty$ conformal systems from classical Toda field theories”, Nucl. Phys. B, 314:3 (1989), 646–686 | DOI | MR

[46] Bilal A., Gervais J.-L., “Systematic construction of conformal theories with higher-spin Virasoro symmetries”, Nucl. Phys. B, 318:3 (1989), 579–630 | DOI | MR

[47] Boyer C. P., Finley J. D., “Killing vectors in self-dual Euclidean Einstein spaces”, J. Math. Phys., 23 (1982), 1126–1130 | DOI | MR | Zbl

[48] Boyer C. P., Plebański J. F., “An infinite hierarchy of conservation laws and nonlinear superposition principles for self-dual Einstein spaces”, J. Math. Phys., 26:2 (1985), 229–234 | DOI | MR | Zbl

[49] Brandt F., “Bäcklund transformations and zero curvature representations of systems of partial differential equations”, J. Math. Phys., 35 (1994), 2463–2484 | DOI | MR | Zbl

[50] Bullough R. K., Dodd R. K., “Bäcklund transformations for the sine-Gordon equations”, Proc. Roy. Soc. London Ser. A, 351:1667 (1976), 499–523 | DOI | MR | Zbl

[51] Bullough R. K., Dodd R. K., “Polynomial conserved densities for the sine-Gordon equations”, Proc. Roy. Soc. London Ser. A, 352 (1977), 481–503 | DOI | MR

[52] Carlet G., Dubrovin B., Zhang Y., The extended Toda hierarchy, arXiv: /nlin.SI/0306060 | MR

[53] Case K. M., Roos A. M., “Sine-Gordon and modified Korteweg–de Vries charges”, J. Math. Phys., 23:3 (1982), 392–395 | DOI | MR

[54] Cieśliński J., “A generalized formula for integrable classes of surfaces in Lie algebras”, J. Math. Phys., 38:8 (1997), 4255–4272 | DOI | MR | Zbl

[55] Dorfman I., Dirac Structures and Integrability of Nonlinear Evolution Equations, Nonlinear Science: Theory and Applications, John Wiley Sons, Chichester, 1993 | MR

[56] Dunajski M., Mason L. J., “Hyper-Kähler hierarchies and their twistor theory”, Comm. Math. Phys., 213 (2000), 641–672 | DOI | MR | Zbl

[57] Fehér L., O'Raifeartaigh L., Ruelle P., Tsutsui I., Wipf A., “On Hamiltonian reductions of the Wess–Zumino–Novikov–Witten theories”, Phys. Rep., 222:1 (1992), 1–64 | DOI | MR

[58] Gervais J.-L., Matsuo Y., “$W$-geometries”, Phys. Lett. B, 274 (1992), 309–316 | DOI | MR

[59] Gervais J.-L., Matsuo Y., “Classical $A_n$-$W$-geometries”, Comm. Math. Phys., 152 (1993), 317–368 | DOI | MR | Zbl

[60] Gervais J.-L., Saveliev M. V., “$W$-geometry of the Toda systems associated with non-exceptional Lie algebras”, Comm. Math. Phys., 180:2 (1996), 265–296 | DOI | MR | Zbl

[61] Geurts M. L., Martini R., Post G. F., “Symmetries of the WDVV equation”, Acta Appl. Math., 72:1–2 (2002), 67–75 | DOI | MR | Zbl

[62] Gusyatnikova V. N., Samokhin A. V., Titov V. S. et al., “Symmetries and conservation laws of Kadomtsev–Pogutse equations”, Acta Appl. Math., 15:1 (1989), 23–64 | DOI | MR | Zbl

[63] Igonin S., Krasil'shchik I. S., “On one-parametric families of Bäcklund transformations”, Advanced Studies in Pure Mathematics, 37 (2003), 99–114 | MR

[64] Kac V. G., Raina A. K., Bombai Lectures on Highest Weight Representation of Infinite Dimensional Lie Algebras, World Scientific, Singapore, 1987 | MR

[65] Kaliappan P., Lakshmanan M., “Connection between the infinite sequence of Lie–Bäcklund symmetries of the Korteweg–de Vries and sine-Gordon equations”, J. Math. Phys., 23:3 (1982), 456–459 | DOI | MR | Zbl

[66] Kazdan J. L., Warner F. W., “Curvature functions for open $2$-manifolds”, Ann. of Math. (2), 99:2 (1974), 203–219 | DOI | MR | Zbl

[67] Kersten P., Krasil'shchik I., Verbovetsky A., “Hamiltonian operators and $\ell^*$-coverings”, J. Geom. Phys., 50:1–4, 273–302 (2004) | MR

[68] Kiselev A. V., “On the geometry of Liouville equation: symmetries, conservation laws, and Bäcklund transformations”, Acta Appl. Math., 72:1–2 (2002), 33–49 | DOI | MR | Zbl

[69] Kiselev A. V., “On homotopy Lie algebra structures in the rings of differential operators”, Note Mat., 22:1–2 (2003) | MR | Zbl

[70] Kiselev A. V., “On conservation laws for the Toda equations”, Acta Appl. Math., 83:1–2, 175–182 (2004) | MR

[71] Kiselev A. V., Golovko V. A., “Non-abelian coverings over the Liouville equation”, Acta Appl. Math., 83:1–2 (2004), 25–37 | DOI | MR | Zbl

[72] Kiselev A. V., Ovchinnikov A. V., “On the Hamiltonian hierarchies, associated with the hyperbolic Euler equations”, J. Dynam. Control Systems, 10:3 (2004), 431–451 | DOI | MR | Zbl

[73] Krasil'shchik I., A simple method to prove locality of symmetry hierarchies, Preprint DIPS-9/2002, 2002

[74] Krasil'shchik I. S., Kersten P. H. M., Symmetries and Recursion Operators for Classical and Supersymmetric Differential Equations, Kluwer Acad. Publ., Dordrecht, 2000 | MR

[75] Krasil'shchik I., Verbovetsky A., Homological Methods in Equations of Mathematical Physics, Advanced Texts in Mathematics, Open Education and Sciences, Opava, 1998; arXiv: /math.DG/9808130

[76] Leznov A. N., Saveliev M. V., “Spherically symmetric equations in gauge theories for an arbitrary semisimple compact Lie group”, Phys. Lett. B, 79:3 (1978), 294–296 | DOI | MR

[77] Liouville J., “Sur l'equation aux différences partielles $d^2\log\lambda/du\,dv \pm\lambda/(2a^2)=0$”, J. de Math. Pure et Appliquée, 18:1 (1853), 71–72

[78] Magri F., “A simple model of the integrable equation”, J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[79] Martina L., Sheftel M. B., Winternitz P., “Group foliation and non-invariant solutions of the heavenly equation”, J. Phys. A, 34 (2001), 9243–9263 | DOI | MR | Zbl

[80] Marvan M., “Another look on recursion operators”, Proc. Conf. Differential Geometry and Applications, Masaryk Univ., Brno, Czech Republic, 1995, 393–402 | MR

[81] Marvan M., Jets. A software for diferential calculus on jet spaces and diffieties, ver. 4.9 (December 2003) for Maple V. Release 4, arXiv: /diffiety.ac.ru/soft/soft.htmdiffiety.ac.ru/soft/soft.htm | MR

[82] Marvan M., “On the horizontal gauge cohomology and nonremovability of the spectral parameter”, Acta Appl. Math., 72:1–2, 51–65 (2002) | MR | Zbl

[83] Miura R. M., “Korteweg–de Vries equation and generalizations, I”, J. Math. Phys., 9:8 (1968), 1202–1204 | DOI | MR | Zbl

[84] Ovchinnikov A., “Toda systems and $W$-algebras”, Proc. 1st Non-Orthodox School on Nonlinearity and Geometry, ed. D. Wójcik, J. Cieśliński, Polish Sci. Publ. PWN, Warszawa, 1998, 348–358 | MR

[85] Poincaré H., “Les fonctions fuchsiennes et l'equation $\Delta u=\exp(u)$”, J. de Math. Pure et Appliquée., $5^{\mathrm e}$ ser., 1898, no. 4, 157–230

[86] Razumov A. V., Saveliev M. V., Lie Algebras, Geometry, and Toda-Type Systems, Cambridge Lecture Notes in Physics, 8, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[87] Rogers C., Shadwick W. F., Bäcklund Transformations and Their Applications, Academic press, New York, 1982 | MR

[88] Sakovich S. Yu., “On special Bäcklund autotransformations”, J. Phys. A, 24 (1991), 401–405 | DOI | MR | Zbl

[89] Sakovich S. Yu., “On conservation laws and zero-curvature representations of the Liouville equation”, J. Phys. A, 27 (1994), L125–L129 | DOI | MR | Zbl

[90] Saveliev M. V., Vershik A. M., “On the continuous Lie algebras and the Cartan operators”, Comm. Math. Phys., 126 (1989), 367–381 | DOI | MR

[91] Shabat A. B., “Higher symmetries of two-dimensional lattices”, Phys. Lett. A, 200 (1995), 121–133 | DOI | MR | Zbl

[92] Shadwick W. F., “The Bäcklund problem for the equation $\mathrm{dd}^2z/\mathrm{dd}x^1\mathrm{dd}x^2=f(z)$”, J. Math. Phys., 19:11 (1978), 2312–2317 | DOI | MR | Zbl

[93] Sukhorukov A. A., Akhmediev N. N., Intensity Limits for Stationary and Interacting Multi-Soliton Complexes, , 2001 arXiv: /nlin.PS/0103026 | MR

[94] Vinogradov A. M., “The $\mathrm{cC}$-spectral sequence, Lagrangian formalism, and conservation laws. I: The linear theory. II: The nonlinear theory”, J. Math. Anal. Appl., 100:1 (1984), 1–129 | DOI | MR

[95] Wahlquist H. D., Estabrook F. B., “Bäcklund transformation for solutions of the Korteweg–de Vries equation”, Phys. Rev. Lett., 31:23 (1973), 1386–1390 | DOI | MR

[96] Wang J. P., Symmetries and Conservation Laws of Evolution Equations, PhD thesis, Vrije Universiteit, Amsterdam, 1998

[97] Witten E., “Some exact multipseudoparticle solutions of classical Yang–Mills theory”, Phys. Rev. Lett., 38:3 (1977), 121–124 | DOI