zbMATH — the first resource for mathematics

Nonlinear equations and elliptic curves. (English. Russian original) Zbl 0595.35087
J. Sov. Math. 28, 51-90 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 23, 79-136 (1983).
This paper deals with the algebro-geometric or finite zone solution of noninear equations which can be written as zero curvature equation \(U_ t-V_ x+[U,V]=0\), where U(x,t,\(\lambda)\) and V(x,t,\(\lambda)\) are two matrices and the parameter \(\lambda\) is defined on elliptic curves. The author presents the main idea of global finite-zone integration [see the author, Usp. Mat. Nauk 32, No.6(198), 183-208 (1977; Zbl 0372.35002)] and gives a detailed analysis of applications of this technique to some problem based on the theory of elliptic functions.
The papers divided into three chapters. The first chapter gives a good exposition of the relevant notions and results. The second chapter is concerned with the algebro-geometric spectral theory of the Schrödinger difference operator \[ L\psi_ n=c_ n\psi_{n+1}+V_ n\psi_ n+c_{n-1}\psi_{n-1} \] with periodic coefficients \(c_ n=c_{n+N}\), \(v_ n=v_{n+N}\). The third chapter discusses the Peierls model describing the self-consistent behavior of atoms of a lattice. The author gives a necessary and sufficient condition for a functional related to the model to be extremal on some set and the stability of the extremal. This is a well-written paper, it contains good analysis to a number of typical integrable systems such as the KdV, sine-Gordon, Landau-Lifshits equations, the principal chiral model etc.
Reviewer: G.-Z.Tu

35Q99 Partial differential equations of mathematical physics and other areas of application
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
33E05 Elliptic functions and integrals
Full Text: DOI
[1] N. I. Akhiezer, ”A continual analogue of orthogonal polynomials on a system of intervals,” Dokl. Akad. Nauk SSSR,141, No. 2, 263–266 (1961).
[2] H. Bateman and E. Erdelyi, Higher Transcendental Functions [in Russian], Nauka, Moscow (1974).
[3] E. D. Belokolos, ”The Peierls-Frolich problems and finite-zone potentials. I,” Teor. Mat. Fiz.,45, No. 2, 268–280 (1980).
[4] E. D. Belokolos, ”The Peierls-Frölich problems and finite-zone potentials. II,” Teor. Mat. Fiz.,48, No. 1, 60–69 (1981).
[5] S. A. Brazovskii, S. A. Gordyunin, and N. N. Kirova, ”Exact solution of the Peierls model with an arbitrary number of electrons on an elementary cell,” Pis’ma Zh. Eksp. Teor. Fiz.,31, No. 8, 486–490 (1980).
[6] S. A. Brazovskii, I. E. Dzyaloshinskii, and N. N. Kirova, ”Spin states in the Peierls model and finite-zone potentials,” Zh. Eksp. Teor. Fiz.,82, No. 6, 2279–2298 (1981).
[7] S. A. Brazovskii, I. E. Dzyaloshinskii, and I. M. Krichever, ”Exactly solvable discrete Peierls models,” Zh. Eksp. Toer. Fiz.,83 No. 1, 389–415 (1982).
[8] I. M. Gel’fand and L. A. Dikii, ”Asymptotics of the resolvent of Sturm-Liouville equations and the algebra of Korteweg-de Vries equations,” Usp. Mat. Nauk,30, No. 5, 67–100 (1975).
[9] I. E. Dzyaloshinskii, ”Theory of helicoidal structures,” Zh. Eksp. Teor. Fiz.,47, No. 5, 992–1008 (1964).
[10] I. E. Dzyaloshinskii and I. M. Krichever, ”Effects of commensurability in the discrete Peierls model,” Zh. Eksp. Teor. Fiz.,83, No. 5, 1576–1581 (1982).
[11] I. E. Dzyaloshinskii and I. M. Krichever, ”Sound and the wave of charge density in the discrete Peierls model,” Zh. Eksp. Teor. Fiz.,95 (1983). (in the press)
[12] B. A. Dubrovin, ”The periodic problem for the Korteweg-de Vries equation in the class of finite-zone potentials,” Funkts. Anal. Prilozhen.,9, No. 3, 41–51(1975). · Zbl 0316.30019
[13] B. A. Dubrovin, ”Theta functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 11–80 (1981). · Zbl 0478.58038
[14] B. A. Dubrovin, ”The inverse problem of scattering theory for periodic finite-zone potentials,” Funkts. Anal. Prilozhen.,9, No. 1, 65–66 (1975). · Zbl 0318.34038
[15] B. A. Dubrovin, V. B. Matveev, and S. P. Novikov, ”Nonlinear equations of Korteweg-de Vries type, finite-zone linear operators, and Abelian manifolds,” Usp. Mat. Nauk,31, No. 1, 55–136 (1976). · Zbl 0326.35011
[16] B. A. Dubrovin and S. M. Natanzon, ”Real two-zone solutions of the sine-Gordon equation,” Funkts. Anal. Prilozhen.,16, No. 1, 27–43 (1982). · Zbl 0554.35100
[17] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons: Method of the Inverse Problem [in Russian], Nauka, Moscow (1980). · Zbl 0598.35002
[18] V. E. Zakharov and A. V. Mikhailov, ”Relativistically invariant two-dimensional models of field theory integrable by the method of the inverse problem,” Zh. Eksp. Teor. Fiz.,74, No. 6, 1953–1974 (1978).
[19] V. E. Zakharov and A. B. Shabat, ”Integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem. II,” Funkts. Anal. Prilozhen.,13, No. 3, 13–22 (1979).
[20] V. E. Zakharov and A. B. Shabat, ”The scheme of integration of nonlinear equations of mathematical physics by the method of the inverse scattering problem. I,” Funkts. Anal. Prilozhen.,8, No. 3, 43–53 (1974). · Zbl 0303.35024
[21] E. I. Zverovich, ”Boundary-value problems of the theory of analytic functions,” Usp. Mat. Nauk,26, No. 1, 113–181 (1971). · Zbl 0217.10201
[22] A. R. Its, ”On finite-zone solutions of equations,” see: V. B. Matveev, Abelian Functions and Solitons, Preprint of Wroclaw Univ., No. 373 (1976).
[23] A. R. Its and V. B. Matveev, ”On a class of solutions of the Korteweg-de Vries equation,” in: Probl. Mat. Fiz., No. 8, Leningrad Univ. (1976), pp. 70–92.
[24] V. A. Kozel and V. P. Kotlyarov, ”Almost-periodic solutions of the equation uttxx+ sin u=0,” Dokl. Akad. Nauk Ukr. SSR,A, No. 10, 878–881 (1976). · Zbl 0337.35003
[25] I. M. Krichever, ”Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 180–208 (1977). · Zbl 0372.35002
[26] I. M. Krichever, ”Integration of nonlinear equations by methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977). · Zbl 0346.35028
[27] I. M. Krichever, ”Commutative rings of ordinary linear differential operators,” Funkts. Anal. Prilozhen.,12, No. 3, 20–31 (1978). · Zbl 0405.58049
[28] I. M. Krichever, ”An analogue of the D’Alembert formula for equations of the principal chiral field and the sine-Gordon equation,” Dokl. Akad. Nauk SSSR,253, No. 2, 288–292 (1980).
[29] I. M. Krichever, ”The Peierls model,” Funkts. Anal. Prilozhen.,16, No. 4, 10–26 (1982). · Zbl 0508.58020
[30] I. M. Krichever, ”Algebrogeometric spectral theory of the Schrödinger difference operator and the Peierls model,” Dokl. Akad. Nauk SSSR,265, No. 5, 1054–1058 (1982).
[31] I. M. Krichever, ”On rational solutions of the Kadomtsev-Petviashvili equation and on integrable systems of particles on the line,” Funkts. Anal. Prilozhen.,12, No. 1, 76–78 (1978). · Zbl 0374.70008
[32] I. M. Krichever, ”Algebrogeometric construction of the Zakharov-Shabat equations and their periodic solutions,” Dokl. Akad. Nauk SSSR,227, No. 2, 291–294 (1976).
[33] I. M. Krichever, ”Algebraic curves and nonlinear difference equations,” Usp. Mat. Nauk,33, No. 4, 215–216 (1978). · Zbl 0382.39003
[34] I. M. Krichever, ”Elliptic solutions of the Kadomtsev-Petviashvili equation and integrable systems of particles,” Funkts. Anal. Prilozhen.,14, No. 4, 45–54 (1980). · Zbl 0455.14011
[35] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles over algebraic curves and non-linear equations,” Usp. Mat. Nauk,35, No. 6, 47–68 (1980). · Zbl 0501.35071
[36] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles and nonlinear equations. Finitezone solutions of rank 2,” Dokl. Akad. Nauk SSSR,247, No. 1, 33–37 (1979).
[37] I. M. Krichever and S. P. Novikov, ”Holomorphic bundles over Riemann surfaces and the Kadomtsev-Petviashvili (KP) equation. I,” Funkts. Anal. Prilozhen.,12, No. 4, 41–52 (1978). · Zbl 0393.35061
[38] S. V. Manakov, ”On complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, No. 2, 543–555 (1974).
[39] N. I. Muskhelishvili, Singular Integral Equations [in Russian], Fizmatgiz, Moscow (1962). · Zbl 0103.07502
[40] S. P. Novikov, ”The periodic problem for the Korteweg-de Vries equation,” Funkts. Anal. Prilozhen.,8, No. 3, 54–66 (1974). · Zbl 0301.54027
[41] R. Peierls, Quantum Theory of the Solid State [Russian translation], IL, Moscow (1956).
[42] J. Serre, Algebraic Groups and Class Fields [Russian translation], Mir, Moscow (1968).
[43] E. Scott (ed.), Solitons in Action [Russian translation], Mir, Moscow (1981).
[44] G. Springer, Introduction to the Theory of Riemann Surfaces [Russian translation], IL, Moscow (1961).
[45] I. V. Cherednik, ”Algebraic aspects of two-dimensional chiral fields. I,” in: Sov. Probl. Mat. (Itogi Nauki i Tekhniki VINITI AN SSSR),17, Moscow (1981), pp. 175–218.
[46] I. V. Cherednik, ”On realness conditions in ’finite-zone integration,”’ Dokl. Akad. Nauk SSSR,252, No. 5, 1104–1108 (1980).
[47] I. V. Cherednik, ”On integrability of a two-dimensional asymmetrical chiralO(3)-field and its quantum analogue,” Yad. Fiz.,33, No. 1, 278–281 (1981).
[48] I. V. Cherednik, ”On solutions of algebraic type of asymmetric differential equations,” Funkts. Anal. Prilozhen.,15, No. 3, 93–94 (1981). · Zbl 0505.35071
[49] M. A. Ablowitz, D. J. Kaup, A. S. Newell, and H. Segur, ”Method for solving the sine-Gordon equation,” Phys. Rev. Lett.,30, 1262–1264 (1973).
[50] S. Aubry, ”Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Israel Phys. Soc.,3, 133–164 (1980). · Zbl 0943.82510
[51] S. Aubry, ”Metal-insulator transition in one-dimensional deformable lattices,” Bifurcation Phenomena in Math. Phys. and Related Topics, C. Bardos and D. Bessis (eds.) (1980), pp. 163–184.
[52] H. M. Baker, ”Note on the foregoing paper ’Commutative ordinary differential operators,”’ Proc. R. Soc. London,118, 584–593 (1928). · JFM 54.0439.02
[53] R. K. Bullough and P. J. Caudrey (eds.), Solitons, Springer-Verlag (1980). · Zbl 0428.00010
[54] F. Calodgero, ”Exactly solvable one-dimensional manybody systems,” Lett. Nuovo Cimento,13, 411–415 (1975).
[55] D. V. Choodnovsky and G. V. Choodnovsky, ”Pole expansions of nonlinear partial differential equations,” Lett. Nuovo Cimento,40B, 339–350 (1977). · Zbl 0324.02066
[56] E. I. Dinaburg and Y. C. Sinai, ”Schrödinger equation with quasiperiodic potentials,” Fund. Anal.,9, 279–283 (1976). · Zbl 0333.34014
[57] L. D. Faddeev, Quantum Scattering Transformation, Proc. Freiburg Summer Inst., 1981, Plenum Press (1982).
[58] H. Flaschka, ”Toda lattice. II,” Prog. Theor. Phys.,51, 543–555 (1974). · Zbl 0942.37505
[59] C. Gardner, J. Green, M. Kruskas, and R. Miura, ”A method for solving the Korteweg-de Vries equation,” Phys. Rev. Lett.,19, 1095–1098 (1967).
[60] P. D. Lax, ”Integrals of nonlinear equations of evolution and solitary waves,” Commun. Pure Appl. Math.,21, No. 5, 467–490 (1968). · Zbl 0162.41103
[61] A. N. Leznov and M. N. Saveliev, ”On the two-dimensional system of differential equations,” Commun. Math. Phys.,74, 111–119 (1980). · Zbl 0429.35063
[62] P. Mansfild, ”Solutions of the Toda lattice,” Preprint Cambridge Univ., Cambridge, CB 39 EW (1982).
[63] A. V. Mikhailov, ”The reduction problem in the Zakharov-Shabat equations,” Physica 3D,1, 215–243 (1981).
[64] A. V. Mikhailov, ”The Landau-Lifshits equation and the Riemann-Hilbert boundary problem on the torus,” Phys. Lett.,92a, 2, 51–55 (1982).
[65] A. M. Perelomov, ”Completely integrable classical systems connected with semisimple Lie algebras,” Lett. Math. Phys.,1, 531–540 (1977).
[66] K. Pohlmeyer, ”Integrable Hamiltonian systems and interaction through quadratic constraints,” Commun. Math. Phys.,46, 207–223 (1976). · Zbl 0996.37504
[67] E. K. Sklyanin, ”On complete integrability of the Landau-Lifshitz equation,” Preprint LOMI E-3-1979, Leningrad (1979). · Zbl 0449.35089
[68] W. P. Su, I. R. Schriffer, and A. I. Heeger, ”Soliton excitations in polyacetylene,” Phys. Rev.,B22, 2099–2108 (1980).
[69] M. Toda, ”Waves in nonlinear lattices,” Prog. Theor. Phys. Suppl.,45, 174–200 (1970).
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.