# Nonlinear equations and elliptic curves

Article

- 100 Downloads
- 9 Citations

## Abstract

The main ideas of global “finite-zone integration” are presented, and a detailed analysis is given of applications of the technique developed to some problems based on the theory of elliptic functions. In the work the Peierls model is integrated as an important application of the algebrogeometric spectral theory of difference operators.

## Keywords

Detailed Analysis Main Idea Nonlinear Equation Difference Operator Important Application
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Literature cited

- 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).Google Scholar
- 2.H. Bateman and E. Erdelyi, Higher Transcendental Functions [in Russian], Nauka, Moscow (1974).Google Scholar
- 3.E. D. Belokolos, “The Peierls-Frolich problems and finite-zone potentials. I,” Teor. Mat. Fiz.,45, No. 2, 268–280 (1980).Google Scholar
- 4.E. D. Belokolos, “The Peierls-Frölich problems and finite-zone potentials. II,” Teor. Mat. Fiz.,48, No. 1, 60–69 (1981).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 9.I. E. Dzyaloshinskii, “Theory of helicoidal structures,” Zh. Eksp. Teor. Fiz.,47, No. 5, 992–1008 (1964).Google Scholar
- 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).Google Scholar
- 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)Google Scholar
- 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).Google Scholar
- 13.B. A. Dubrovin, “Theta functions and nonlinear equations,” Usp. Mat. Nauk,36, No. 2, 11–80 (1981).Google Scholar
- 14.B. A. Dubrovin, “The inverse problem of scattering theory for periodic finite-zone potentials,” Funkts. Anal. Prilozhen.,9, No. 1, 65–66 (1975).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 21.E. I. Zverovich, “Boundary-value problems of the theory of analytic functions,” Usp. Mat. Nauk,26, No. 1, 113–181 (1971).Google Scholar
- 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).Google Scholar
- 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.Google Scholar
- 24.V. A. Kozel and V. P. Kotlyarov, “Almost-periodic solutions of the equation u
_{tt}−u_{xx}+ sin u=0,” Dokl. Akad. Nauk Ukr. SSR,A, No. 10, 878–881 (1976).Google Scholar - 25.I. M. Krichever, “Methods of algebraic geometry in the theory of nonlinear equations,” Usp. Mat. Nauk,32, No. 6, 180–208 (1977).Google Scholar
- 26.I. M. Krichever, “Integration of nonlinear equations by methods of algebraic geometry,” Funkts. Anal. Prilozhen.,11, No. 1, 15–31 (1977).Google Scholar
- 27.I. M. Krichever, “Commutative rings of ordinary linear differential operators,” Funkts. Anal. Prilozhen.,12, No. 3, 20–31 (1978).Google Scholar
- 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).Google Scholar
- 29.I. M. Krichever, “The Peierls model,” Funkts. Anal. Prilozhen.,16, No. 4, 10–26 (1982).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 33.I. M. Krichever, “Algebraic curves and nonlinear difference equations,” Usp. Mat. Nauk,33, No. 4, 215–216 (1978).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 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).Google Scholar
- 38.S. V. Manakov, “On complete integrability and stochastization in discrete dynamical systems,” Zh. Eksp. Teor. Fiz.,67, No. 2, 543–555 (1974).Google Scholar
- 39.N. I. Muskhelishvili, Singular Integral Equations [in Russian], Fizmatgiz, Moscow (1962).Google Scholar
- 40.S. P. Novikov, “The periodic problem for the Korteweg-de Vries equation,” Funkts. Anal. Prilozhen.,8, No. 3, 54–66 (1974).Google Scholar
- 41.R. Peierls, Quantum Theory of the Solid State [Russian translation], IL, Moscow (1956).Google Scholar
- 42.J. Serre, Algebraic Groups and Class Fields [Russian translation], Mir, Moscow (1968).Google Scholar
- 43.E. Scott (ed.), Solitons in Action [Russian translation], Mir, Moscow (1981).Google Scholar
- 44.G. Springer, Introduction to the Theory of Riemann Surfaces [Russian translation], IL, Moscow (1961).Google Scholar
- 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.Google Scholar
- 46.I. V. Cherednik, “On realness conditions in ‘finite-zone integration,’” Dokl. Akad. Nauk SSSR,252, No. 5, 1104–1108 (1980).Google Scholar
- 47.I. V. Cherednik, “On integrability of a two-dimensional asymmetrical chiral
*O*(3)-field and its quantum analogue,” Yad. Fiz.,33, No. 1, 278–281 (1981).Google Scholar - 48.I. V. Cherednik, “On solutions of algebraic type of asymmetric differential equations,” Funkts. Anal. Prilozhen.,15, No. 3, 93–94 (1981).Google Scholar
- 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).CrossRefGoogle Scholar
- 50.S. Aubry, “Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Israel Phys. Soc.,3, 133–164 (1980).Google Scholar
- 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.Google Scholar
- 52.H. M. Baker, “Note on the foregoing paper ‘Commutative ordinary differential operators,’” Proc. R. Soc. London,118, 584–593 (1928).Google Scholar
- 53.R. K. Bullough and P. J. Caudrey (eds.), Solitons, Springer-Verlag (1980).Google Scholar
- 54.F. Calodgero, “Exactly solvable one-dimensional manybody systems,” Lett. Nuovo Cimento,13, 411–415 (1975).Google Scholar
- 55.D. V. Choodnovsky and G. V. Choodnovsky, “Pole expansions of nonlinear partial differential equations,” Lett. Nuovo Cimento,40B, 339–350 (1977).Google Scholar
- 56.E. I. Dinaburg and Y. C. Sinai, “Schrödinger equation with quasiperiodic potentials,” Fund. Anal.,9, 279–283 (1976).CrossRefGoogle Scholar
- 57.L. D. Faddeev, Quantum Scattering Transformation, Proc. Freiburg Summer Inst., 1981, Plenum Press (1982).Google Scholar
- 58.H. Flaschka, “Toda lattice. II,” Prog. Theor. Phys.,51, 543–555 (1974).Google Scholar
- 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).CrossRefGoogle Scholar
- 60.P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Commun. Pure Appl. Math.,21, No. 5, 467–490 (1968).Google Scholar
- 61.A. N. Leznov and M. N. Saveliev, “On the two-dimensional system of differential equations,” Commun. Math. Phys.,74, 111–119 (1980).CrossRefGoogle Scholar
- 62.P. Mansfild, “Solutions of the Toda lattice,” Preprint Cambridge Univ., Cambridge, CB 39 EW (1982).Google Scholar
- 63.A. V. Mikhailov, “The reduction problem in the Zakharov-Shabat equations,” Physica 3D,1, 215–243 (1981).Google Scholar
- 64.A. V. Mikhailov, “The Landau-Lifshits equation and the Riemann-Hilbert boundary problem on the torus,” Phys. Lett.,92a, 2, 51–55 (1982).Google Scholar
- 65.A. M. Perelomov, “Completely integrable classical systems connected with semisimple Lie algebras,” Lett. Math. Phys.,1, 531–540 (1977).CrossRefGoogle Scholar
- 66.K. Pohlmeyer, “Integrable Hamiltonian systems and interaction through quadratic constraints,” Commun. Math. Phys.,46, 207–223 (1976).CrossRefGoogle Scholar
- 67.E. K. Sklyanin, “On complete integrability of the Landau-Lifshitz equation,” Preprint LOMI E-3-1979, Leningrad (1979).Google Scholar
- 68.W. P. Su, I. R. Schriffer, and A. I. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev.,B22, 2099–2108 (1980).Google Scholar
- 69.M. Toda, “Waves in nonlinear lattices,” Prog. Theor. Phys. Suppl.,45, 174–200 (1970).Google Scholar

## Copyright information

© Plenum Publishing Corporation 1985