×

Nonlinear evolution equations and hyperelliptic covers of elliptic curves. (English) Zbl 1254.14041

A huge variety of nonlinear integrable processes and phenomena in physics and mathematics can be described by a few nonlinear partial derivative equations: Korteweg-deVries (KdV) and Kadomtsev-Petviashvili (KP), 1D and 2D Toda, sine-Gordon, nonlinear Schrödinger. The present paper is a further contribution to the study of exact solutions to these equations. The author will be concerned with algebro-geometric solutions, doubly periodic in one variable. According to the well known Its-Matveev’s formulae, the Jacobians of the corresponding spectral curves must contain an elliptic curve \(X\), satisfying suitable geometric properties. It turns out that the latter curves are in fact contained in a particular algebraic surface \(S^\bot\), projecting onto a rational surface \(\widetilde{S}\). Moreover, all spectral curves project onto a rational curve inside. The author is thus led to study all rational curves of \(\widetilde{S}\), having suitable numerical equivalence classes. At last he obtains \(d-1\)-dimensional of spectral curves, of arbitrary high genus, giving rise to KdV solutions doubly periodic with respect to the \(d\)-th KdV flow (\(d \geq 1\)). A completely analogous constructive approach can be worked out for the other three cases. Some results are presented, without proof, for the 1D Toda, nonlinear Schrödinger and sine-Gordon equations.

MSC:

14H70 Relationships between algebraic curves and integrable systems
14H81 Relationships between algebraic curves and physics
35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
14H30 Coverings of curves, fundamental group
14H40 Jacobians, Prym varieties
14H55 Riemann surfaces; Weierstrass points; gap sequences
14C20 Divisors, linear systems, invertible sheaves
35C08 Soliton solutions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Airault, H., McKean, H.P., and Moser, J., Rational and Elliptic Solutions of the Korteweg-de Vries Equation and a Related Many Body Problem, Comm. Pure Appl. Math., 1977, vol. 30, pp. 95–148. · Zbl 0338.35024 · doi:10.1002/cpa.3160300106
[2] Akhmetshin, A.A., Krichever, I. M., and Vol’vovskii, Y. S., Elliptic Families of Solutions of the Kadomtsev-Petviashvili Equation, and the Field Analog of the Elliptic Calogero-Moser System, Funktsional. Anal. i Prilozhen., 2002, vol. 36, no. 4, pp. 1–17 [Funct. Anal. Appl., 2002, vol. 36, no. 4, pp. 253–266]. · doi:10.4213/faa215
[3] Babich, M.V., Bobenko, A. I., and Matveev, V.B., Solutions of Nonlinear Equations Integrable in Jacobi Theta Functions by the Method of the Inverse Problem, and Symmetries of Algebraic Curves, Izv. Akad. Nauk SSSR Ser. Mat., 1985, vol. 49, no. 3, pp. 511–529 [Math. USSR Izv., 1986, vol. 26, no. 3, pp. 479–496]. · Zbl 0583.35012
[4] Belokolos, E. D., Bobenko, A. I., Ènol’skii, V. Z., and Matveev, V. B., Algebraic-Geometric Principles of Superposition of Finite-Zone Solutions of the Integrable Nonlinear Equations, Uspekhi Math. Nauk, 1986, vol. 41, no. 2(248), pp. 3–42 [Russian Math. Surveys, 1986, vol. 41, no. 2, pp. 1–49].
[5] Bobenko, A. I., Periodic Finite-Zone Solutions of the Sine-Gordon Equation, Funktsional. Anal. i Prilozhen., 1984, vol. 18, no. 3, pp. 73–74 [Funct. Anal. Appl., 1984, vol. 18, no. 3, pp. 240–242]. · Zbl 0537.06009 · doi:10.1007/BF01076373
[6] Calogero, F., Solution of the One Dimensional n-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials, J. Math. Phys., 1971, vol. 12, pp. 419–436 (see also erratum to this paper: J. Math. Phys., 1996, vol. 37, no. 7, p. 3646). · doi:10.1063/1.1665604
[7] Dubrovin, B.A., Matveev, V.B., and Novikov, S.P., Nonlinear Equations of KdV Type, Finite-Zone Linear Operators and Abelian Varieties, Uspekhi Math. Nauk, 1976, vol. 31, no. 1(187), pp. 55–136 [Russian Math. Surveys, 1976, vol. 31, no. 1, pp. 59–146]. · Zbl 0326.35011
[8] Dubrovin, B.A. and Natanzon, S.M., Real Two-Zone Solutions of the Sine-Gordon Equation, Funktsional. Anal. i Prilozhen., 1982, vol. 16, no. 1, pp. 27–43 [Funct. Anal. Appl., 1982, vol. 16, no. 1, pp. 21–33]. · Zbl 0554.35100 · doi:10.1007/BF01081805
[9] Dubrovin, B.A. and Novikov, S.P., A Periodicity Problem for the Korteweg-de Vries and Sturm-Liouville Equations: Their Connection with Algebraic Geometry, Dokl. Acad. Nauk SSSR, 1974, vol. 219, no. 3, pp. 531–534 [Sov. Math. Dokl., 1974, vol. 15, pp. 1597–1601]. · Zbl 0312.35015
[10] Flédrich, P., Paires 3-tangentielles hyperelliptiques et solutions doublement périodiques en t de l’équation de Korteweg-de Vries, Thèse, Université d’Artois (Pôle de Lens), Déc. 2003.
[11] Flédrich, P. and Treibich, A., Hyperelliptic Osculating Covers and KdV Solutions Periodic in t, Int. Math. Res. Not., 2006, vol. 5, Art. ID 73476 (17 pp.). · Zbl 1142.35077
[12] Hartshorne, R., Algebraic Geometry, Grad. Texts in Math., vol. 52, New York-Heidelberg: Springer, 1977. · Zbl 0367.14001
[13] Hirota, R., Direct Methods of Finding Exact Solutions of Non-Linear Evolution Equations, in Bäcklund Transformations, the Inverse Scattering Method, and Their Applications, R. M. Miura (Ed.), Lecture Notes in Math., vol. 515, Berlin-New York: Springer, 1976, pp. 40–68.
[14] Its, A.R. and Matveev, V. B., Schrödinger Operators with Finite-Gap Spectrum and N-soliton Solutions of the Korteweg-de Vries Equation, Teoret. Mat. Fiz., 1975, vol. 23, no. 1, pp. 51–68 [Theoret. and Math. Phys., vol. 23, no. 1, pp. 343–355].
[15] Kashiwara, M. and Miwa, T., Transformation Groups for Soliton Equations: 1. The {\(\tau\)}-Function of the Kadomtsev-Petviashvili Equation, Proc. Japan Acad. Ser. A Math. Sci., 1981, vol. 57, pp. 342–347. · Zbl 0538.35065 · doi:10.3792/pjaa.57.342
[16] Kozel, V.A. and Kotlyarov, V.P., Almost Periodic Solutions of the Equation u tt -u xx +sin u = 0, Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1976, no. 10, pp. 878–881, 959 (Russian). · Zbl 0337.35003
[17] Krichever, I.M., Integration of Nonlinear Equations by the Methods of Algebraic Geometry, Funktsional. Anal. i Prilozhen., 1977, vol. 11, no. 1, pp. 15–31 [Funct. Anal. Appl., 1977, vol. 11, no. 1, pp. 12–26]. · Zbl 0368.35022 · doi:10.1007/BF01135528
[18] Krichever, I. M., Elliptic Solutions of the Kadomtsev-Petviashvili Equation and Integrable Systems of Particles, Funktsional. Anal. i Prilozhen., 1980, vol. 14, no. 4, pp. 45–54 [Funct. Anal. Appl., 1980, vol. 14, no. 4, pp. 282–290]. · Zbl 0454.35078 · doi:10.1007/BF01086193
[19] Kruskal, M.D., The Korteweg-de Vries Equation and Related Evolution Equations, in Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, NY, 1972), Lectures in Appl. Math., vol. 15, Providence, RI: AMS, 1974, pp. 61–83.
[20] Lamb, G. L., Elements of Soliton Theory, New York: Wiley, 1980. · Zbl 0445.35001
[21] Lax, P. D., Periodic Solutions of the KdV Equation, Comm. Pure Appl. Math., 1975, vol. 28, pp. 141–188. · Zbl 0295.35004 · doi:10.1002/cpa.3160280105
[22] McKean, H.P. and van Moerbeke, P., The Spectrum of Hill’s Equation, Invent. Math., 1975, vol. 30, no. 3, pp. 217–274. · Zbl 0319.34024 · doi:10.1007/BF01425567
[23] Mumford, D., An Algebro-Geometric Construction of Commuting Operators and of Solutions to the Toda Lattice Equation, Korteweg-de Vries Equation and Related Nonlinear Equations, in Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977), Tokyo: Kinokuniya Book Store, 1978, pp. 115–153. · Zbl 0423.14007
[24] Previato, E., Hyperelliptic Quasi-Periodic and Soliton Solutions of the Nonlinear Schrödinger Equation, Duke Math. J., 1985, vol. 52, no. 2, pp. 329–377. · Zbl 0578.35086 · doi:10.1215/S0012-7094-85-05218-4
[25] Sato, M. and Sato, Y., Soliton Equations as Dynamical Systems on Infinite Dimensional Grassmann Manifold, in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), H. Fujita, P.D. Lax, and G. Strang (Eds.), North-Holland Math. Stud., vol. 81, Amsterdam: North-Holland, 1983, pp. 259–271. · Zbl 0528.58020
[26] Segal, G. and Wilson, G., Loop Groups and Equations of KdV Type, Publ. Math. Inst. Hautes Études Sci., 1985, vol. 61, pp. 5–65. · Zbl 0592.35112 · doi:10.1007/BF02698802
[27] Smirnov, A.O., Elliptic Solutions of the Korteweg-de Vries Equation, Mat. Zametki, 1989, vol. 45, no. 6, pp. 66–73 [Math. Notes, 1989, vol. 45, no. 6, pp. 467–481].
[28] Smirnov, A.O., Finite-Gap Solutions of Abelian Toda Chain of Genus 4 and 5 in Elliptic Functions, Teoret. Mat. Fiz., 1989, vol. 78, no. 1, pp. 11–21 [Theoret. and Math. Phys., 1989, vol. 78, no. 1, pp. 6–13].
[29] Smirnov, A.O., Solutions of the KdV Equation Elliptic in t, Teoret. Mat. Fiz., 1994, vol. 100, no. 2, pp. 183–198 [Theoret. and Math. Phys., 1994, vol. 100, no. 2, pp. 937–947].
[30] Smirnov, A.O., Elliptic Solution of the Nonlinear Schrödinger Equation and the Modified Korteweg-de Vries Equation, Math. Sb., 1994, vol. 185, no. 8, pp. 103–114 [Sb. Math., 1995, vol. 82, no. 2, pp. 461–470]. · Zbl 0854.35110
[31] Toda, M., Waves in Nonlinear Lattices, Progr. Theoret. Phys. Suppl., 1970, vol. 45, pp. 174–200. · doi:10.1143/PTPS.45.174
[32] Treibich, A., Tangential Polynomials and Elliptic Solitons, Duke Math. J., 1989, vol. 59, no. 3, pp. 611–627. · Zbl 0698.14029 · doi:10.1215/S0012-7094-89-05928-0
[33] Treibich, A., Matrix Elliptic Solitons, Duke Math. J., 1997, vol. 90, no. 3, pp. 523–547. · Zbl 0909.35116 · doi:10.1215/S0012-7094-97-09014-1
[34] Treibich, A., Revêtements hyperelliptiques d-osculateurs et solitons elliptiques de la hiérarchie KdV, C. R. Acad. Sci. Paris, Sér. 1, 2007, vol. 345, pp. 213–218. · Zbl 1118.14036 · doi:10.1016/j.crma.2007.06.019
[35] Treibich, A. and Verdier, J.-L., Solitons elliptiques (with an Appendix by J.Oesterlé), in The Grothendieck Festschrift: Vol. 3, Progr. Math., vol. 88, Boston, MA: Birkhäuser, 1990, pp. 437–480.
[36] Treibich, A. and Verdier, J.-L., Revétements exceptionnels et sommes de 4 nombres triangulaires, Duke Math. J., 1992, vol. 68, no. 2, pp. 217–236. · Zbl 0806.14013 · doi:10.1215/S0012-7094-92-06809-8
[37] Zakharov, V.E. and Shabat, A. B., A Scheme for Integrating the Nonlinear Equations of Mathematical Physics by the Method of the Inverse Scattering Problem: 1, Funktsional. Anal. i Prilozhen., 1974, vol. 8, no. 3, pp. 43–53 [Funct. Anal. Appl., 1974, vol. 8, no. 3, pp. 226–235].
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.