On analytic families of invariant tori for PDES
Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes (II), Astérisque, no. 297 (2004), pp. 35-65.
@incollection{AST_2004__297__35_0,
     author = {Dubrovin, Boris},
     title = {On analytic families of invariant tori for {PDES}},
     booktitle = {Analyse complexe, syst\`emes dynamiques, sommabilit\'e des s\'eries divergentes et th\'eories galoisiennes (II)},
     editor = {Loday-Richaud Mich\`ele},
     series = {Ast\'erisque},
     pages = {35--65},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {297},
     year = {2004},
     mrnumber = {2135674},
     zbl = {1087.35083},
     language = {en},
     url = {http://www.numdam.org/item/AST_2004__297__35_0/}
}
TY  - CHAP
AU  - Dubrovin, Boris
TI  - On analytic families of invariant tori for PDES
BT  - Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes (II)
AU  - Collectif
ED  - Loday-Richaud Michèle
T3  - Astérisque
PY  - 2004
SP  - 35
EP  - 65
IS  - 297
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/AST_2004__297__35_0/
LA  - en
ID  - AST_2004__297__35_0
ER  - 
%0 Book Section
%A Dubrovin, Boris
%T On analytic families of invariant tori for PDES
%B Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes (II)
%A Collectif
%E Loday-Richaud Michèle
%S Astérisque
%D 2004
%P 35-65
%N 297
%I Société mathématique de France
%U http://www.numdam.org/item/AST_2004__297__35_0/
%G en
%F AST_2004__297__35_0
Dubrovin, Boris. On analytic families of invariant tori for PDES, dans Analyse complexe, systèmes dynamiques, sommabilité des séries divergentes et théories galoisiennes (II), Astérisque, no. 297 (2004), pp. 35-65. http://www.numdam.org/item/AST_2004__297__35_0/

[1] L. Ahlfors & L. Sario - Riemann Surfaces, Princeton Mathematical Series, vol. 26, Princeton University Press, Princeton, N.J., 1960. | MR | Zbl

[2] V. I. Arnold - Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, NY, 1978. | DOI | MR | Zbl

[3] E. Belokolos, A. Bobenko, V. Enolskii, A. Its & V. Matveev - Algebro- Geometric Approach to Nonlinear Integrable Equations, Springer-Verlag, NY, 1994. | Zbl

[4] A. Bobenko & L. Bordag - "Periodic multiphase solutions of the Kadomsev-Petviashvili equation", J. Phys. A 22 (1989), p. 1259-1274. | DOI | MR | Zbl

[5] A. Bobenko, N. Ercolani, H. Knörrer & E. Trubowitz - "Density of heat curves in the moduli space", in Panoramas of mathematics (Warsaw, 1992/1994), Banach Center Publ., vol. 34, Polish Acad. Sci., Warsaw, 1995, p. 19-27. | EuDML | MR | Zbl

[6] J. Bourgain - "On the Cauchy problem for the Kadomtsev-Petviashvili equation", Geom. Funct. Anal. 3 (1993), p. 315-341. | DOI | EuDML | MR | Zbl

[7] W. Craig - Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panoramas & Synthèses, vol. 9, Société Mathématique de France, Paris, 2000. | MR | Zbl

[8] B. Dubrovin - "On S.P. Novikov's conjecture in the theory of v-functions and nonlinear equations of Korteweg-de Vries and Kadomcev-Petviasvili type", Dokl. Akad. Nauk SSSR 251 (1980), no. 3, p. 541-544. | MR | Zbl

[9] B. Dubrovin, "The Kadomtsev-Petviashvili equation and relations between periods of holomorphic differentials on Riemann surfaces". Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 5, p. 1015-1028, 1198. | MR | Zbl

[10] B. Dubrovin, "Theta functions and nonlinear equations", Russian Math. Surveys 36 (1981), p. 11-92. | DOI | MR | Zbl

[11] B. Dubrovin, R. Flickinger & H. Segur - "Three-phase solutions of the Kadomtsev- Petviashvili equation", Stud. Appl. Math. 99 (1997), p. 137-203. | DOI | MR | Zbl

[12] B. Dubrovin, V. Matveev & S. Novikov - "Nonlinear equations of Korteweg-de Vries type, finite-band linear operators and Abelian varieties", Russian Math. Surveys 31 (1976), p. 59-146. | DOI | MR | Zbl

[13] B. Dubrovin & S. Natanzon - "Real theta-function solutions of the Kadomtsev- Petviashvili equation", Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 2, p. 267-286, | MR | Zbl

B. Dubrovin & S. Natanzon - "Real theta-function solutions of the Kadomtsev- Petviashvili equation", translation in Math. USSR-Izv. 32 (1989), p. 269-288. | DOI | MR | Zbl

[14] A. Einstein - "Zum Quantensatz von Sommerfeld und Epstein", Deutsche Physikalische Gesellschaft, Verhandlungen 19 (1917), p. 82-92.

[15] H. Eliasson - "Hamiltonian systems with Poisson commuting integrals", Ph.D. Thesis, KTH Stockholm, 1984.

[16] J. Fay - Theta Functions on Riemann Surfaces, Lect. Notes in Math., vol. 352, Springer-Verlag, Berlin-Heidelberg-New York, 1973. | MR | Zbl

[17] J. Feldman, H. Knörrer & E. Trubowitz - Riemann Surfaces of Infinite Genus, CRM Monograph Series, vol. 20, American Mathematical Society, Providence, RI, 2003. | DOI | MR | Zbl

[18] P. Griffiths & J. Harris - Principles of Algebraic Geometry, Pure and Applied Mathematics, Wiley-Interscience (John Wiley & Sons), New York, 1978. | MR | Zbl

[19] J. Hammack, D. Mccallister, N. Scheffner & H. Segur - "Two-dimensional periodic waves in shallow water. II. Asymmetric waves", J. Fluid Mech. 285 (1995), p. 95-122. | DOI | MR

[20] J. Hammack, N. Scheffner & H. Segur - "Two-dimensional periodic waves in shallow water", J. Fluid Mech. 209 (1989), p. 567-589. | DOI | MR

[21] H. Ito - "Convergence of Birkhoff normal forms for integrable systems", Comment. Math. Helv. 64 (1989), p. 412-461. | DOI | EuDML | MR | Zbl

[22] H. Ito, "Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case", Math. Ann. 292 (1992), p. 411-444. | DOI | EuDML | MR | Zbl

[23] B. Kadomtsev & V. Petviashvili - "On the stability of solitary waves in a weakly dispersing medium", Soviet. Phys. Dokl. 15 (1970), p. 539-541. | Zbl

[24] T. Kappeler & J. Pöschel - KdV & KAM, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 45, Springer-Verlag, Berlin, 2003. | MR | Zbl

[25] I. Krichever - "Algebrogeometric construction of the Zakharov-Shabat equations and their periodic solutions", Dokl. Akad. Nauk SSSR 227 (1976), p. 291-294. | MR | Zbl

[26] I. Krichever, "Methods of algebraic geometry in the theory of nonlinear equations", Russian Math. Surveys 32 (1977), p. 185-213. | DOI | Zbl

[27] I. Krichever, "Spectral theory of two-dimensional periodic operators and its applications", Uspekhi Mat. Nauk 44 (1989), no. 2, p. 121-184, | MR | Zbl

I. Krichever, "Spectral theory of two-dimensional periodic operators and its applications", translation in Russian Math. Surveys 44 (1989), p. 145-225. | DOI | MR | Zbl

[28] S. Kuksin - Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lect. Notes in Math., vol. 155, Springer-Verlag, Berlin, 1993. | MR | Zbl

[29] L. D. Landau, E. M. Lifshitz & L. P. Pitaevsky - Statistical physics. Pt.1. 3rd ed., Course of theoretical physics, vol. 5, Pergamon, London, 1980. | Zbl

[30] P. D. Lax & C. D. Levermore - "The small dispersion limit of the Korteweg-de Vries equation I. II. III", Comm. Pure Appl. Math. 36 (1983), p. 253-290; 571-593; 809-829. | DOI | MR | Zbl

[31] H. Mckean & E. Trubowitz - "Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points", Comm. Pure Appl. Math. 29 (1976), p. 143-226. | DOI | MR | Zbl

[32] A. Nakamura - "A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. Exact two-periodic wave solution", J. Phys. Soc. Japan, 47 (1979), p. 1701-1705. | DOI | MR | Zbl

[33] S. P. Novikov, S. V. Manakov, L. P. Pitaevskii & V. E. Zakharov - Theory of solitons. The inverse scattering method. Consultants Bureau (Plenum), New York, 1984. | MR | Zbl

[34] H. Poincaré - Les méthodes nouvelles de la mécanique céleste, Vol. II, Gauthier-Villars, Paris, 1893. | JFM | MR

[35] M. Schwarz JR. - "Commuting flows and invariant tori: Korteweg-de Vries", Adv. in Math. 89 (1991), p. 192-216. | DOI | MR | Zbl

[36] T. Shiota - "Characterization of Jacobian varieties in terms of soliton equations", Invent. Math. 83 (1986), p. 333-382. | DOI | EuDML | MR | Zbl

[37] G. G. Stokes - "On the theory of oscillatory waves", Camb. Trans. 8 (1847), p. 441-473.

[38] S. Venakides - "The continuum limit of theta functions", Comm. Pure Appl. Math. 42 (1989), p. 711-728. | DOI | MR | Zbl

[39] G. B. Whitham - Linear and Nonlinear Waves. Wiley-Interscience (John Wiley & Sons), New York, 1974. | MR | Zbl

[40] V. E. Zakharov & E. I. Schulman - "Integrability of nonlinear systems and perturbation theory", in What is Integrability?, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991, p. 185-250. | DOI | MR | Zbl