Abstract
The year of 1995 is not merely the centenary of the Korteweg—de Vries equation which we celebrate at this conference. It is also the year of the ‘majority’ of the finite-gap or algebraic—geometrical theory of integration of nonlinear equations — one of the most important components of the branch of modern mathematical physics, which is called the theory of integrable systems or the soliton theory. The main goal of this paper is to present the key points of the finite-gap theory and some of its applications. Part of its applications is directly related to the KdV equation, while a part of them lies beyond the framework of not only this particular equation but the theory of soliton equations in general. Corresponding examples refer to the string theory and topological field theory models. Not aspiring to be exhaustive, they manifest versatility of the methods, the origin of which would be forever related to the magic words: Korteweg—de Vries Equation.
Mathematics Subject Classifications (1991): 58F07.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Dubrovin, B., Matveev, V., and Novikov, S.: Non-linear equations of Korteweg-de Vries type, finite zone linear operators and Abelian varieties, Uspekhi Mat. Nauk 31(1) (1976), 55–136.
Zakharov, V., Manakov, S., Novikov, S., and Pitaevski, L.: Soliton Theory, Nauka, Moscow, 1980.
Lax, P.: Periodic solutions of Korteweg-de Vries equation, Comm. Pure Appl. Math. 28 (1975), 141–188.
McKean, H. and van Moerbeke, P.: The spectrum of Hill’s equation, Invent. Math. 30 (1975), 217–274.
Krichever, I. M: The algebraic-geometrical construction of Zakharov-Shabat equations and their periodic solutions, Doklady Akad. Nauk USSR 227(2) (1976), 291–294.
Krichever, I.: The integration of non-linear equations with the help of algebraic-geometrical methods, Funk. Anal. Pril. 11(1) (1977), 15–31.
Krichever, I.: Spectral theory of two-dimensional periodic operators and its applications, Uspekhi Mat. Nauk 44(2) (1989), 121–184.
Krichever, I.: Averaging method for two-dimensional integrable equations, Funk. Anal. Pril. 22(3) (1988), 37–52.
Feldmann, J., Knorrer, H., and Trubowitz, E.: Riemann surfaces of the infinite genus, Preprint ETH, Zurich.
Shiota, T: Characterization of Jacobian varieties in terms of soliton equations, Invent. Math. 83 (1986), 333–382.
Takebe, T.: Representation theoretical meaning for the initial value problem for the Toda lattice hierarchy: I, Lett. Math. Phys. 21 (1991), 77–84.
Sato, M.: Soliton equations and universal Grassmann manifold, Math. Lect. Notes Ser. 18, Sophia University, Tokyo, 1984.
Dubrovin, B., Krichever, I., and Novikov, S.: The Schrödinger equation in a periodic field and Riemann surfaces, Soviet Dokl. 229(1) (1976), 15–18.
Krichever, I.: The t-function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994), 437–475.
Takasaki, K. and Takebe, K.: SDiff(2) Toda equation hierarchy, tau-function and symmetries, Preprint RIMS-790, Kyoto.
Takebe, K.: Area-preserving diffeomorphisms and nonlinear integrable systems, in: Proceedings of Topological and Geometrical Methods in Field Theory, May 1991, Turku, Finland.
Saveliev, M.: On the integrability problem of the continuous long wave approximation of the Toda lattice, Preprint ENSL, Lyon, 1992.
Zakharov, V.: Benney equations and quasi-classical approximation in the inverse problem method, Funk. Anal. Pril. 14(2) (1980), 89–98.
Tsarev, S.: The geometry of Hamiltonian systems of hydrodynamic type, Izvest. USSR, Ser. Matem. 54(5) (1990), 96–154.
Dubrovin, B. and Novikov, S.: The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and Bogolyubov-Whitham averaging method, Sov. Math. Dokl. 27 (1983) 665–669.
Dubrovin, B. and Novikov, S.: Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory, Russ. Math. Surveys 44(6) (1989), 35–124.
Novikov, S.: The geometry of conservative systems of hydrodynamics type. The method of averaging for field-theoretical systems, Russ. Math Surveys 40(4) (1985), 85–98.
Krichever, I.: The dispersionless Lax equations and topological minimal models, Comm. Math. Phys. 143(2) (1992), 415–429.
Krichever, I.: Whitham theory for integrable systems and topological field theories, to appear in: Proceedings of Summer Cargese School, July, 1991.
Verlinder, E. and Verlinder, H.: A solution of two-dimensional topological quantum gravity, Preprint IASSNS-HEP-90/40, PUPT-1176, 1990.
Dubrovin, B.: Hamiltonian formalism of Whitham-type hierarchies and topological Landau-Ginsburg models, Comm. Math. Phys. 145(1) (1992), 195–207.
Dubrovin, B.: Differential geometry of moduli spaces and its application to soliton equations and to topological conformal field theory, Preprint No. 117, Scuola Normale Superiore, Pisa, November 1991.
Blok, B. and Varchenko, A.: Topological conformal field theories and the flat coordinates, Int. J. Modern Phys. A 7, 1467–1490.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Krichever, I. (1995). Algebraic—Geometrical Methods in the Theory of Integrable Equations and Their Perturbations. In: Hazewinkel, M., Capel, H.W., de Jager, E.M. (eds) KdV ’95. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0017-5_6
Download citation
DOI: https://doi.org/10.1007/978-94-011-0017-5_6
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4011-2
Online ISBN: 978-94-011-0017-5
eBook Packages: Springer Book Archive