Skip to main content
Log in

Algebraic aspects of nonlinear differential equations

  • Published:
Journal of Soviet Mathematics Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Literature cited

  1. O. I. Bogoyavlenskii and S. P. Novikov, “On the connection of the Hamiltonian formalisms of stationary and nonstationary problems,” Funkts. Analiz Prilozhen.,10, No. 1, 9–13 (1976).

    Google Scholar 

  2. I. M. Gel'fand and L. A. Dikii, “Asymptotic resolvents of Sturm-Liouville equations and the algebra of the Korteweg-de Vries equation,” Usp. Mat. Nauk,30, No. 5, 67–100 (1975).

    Google Scholar 

  3. I. M. Gel'fand and L. A. Dikii, “The structure of Lie algebras in the formal calculus of variations,” Funkts. Analiz Prilozhen.,10, No. 1, 28–36 (1976).

    Google Scholar 

  4. I. M. Gel'fand and L. A. Dikii, “Fractional powers of operators and Hamiltonian systems,” Funkts. Analiz Prilozhen.,10, No. 4, 13–29 (1976).

    Google Scholar 

  5. I. M. Gel'fand and L. A. Dikii, “Resolvents and Hamiltonian systems,” Funkts. Analiz Prilozhen.,11, No. 2, 11–27 (1977).

    Google Scholar 

  6. I. M. Gel'fand, Yu. I. Manin, and M. A. Shubin, “Poisson brackets and the kernel of the variational derivative in the formal calculus of variations,” Funkts. Analiz Prilozhen.,10, No. 4, 30–34 (1976).

    Google Scholar 

  7. V. G. Drinfel'd, “On commutative subrings of some noncommutative rings,” Funkts. Analiz Prilozhen.,11, No. 1, 11–14 (1977).

    Google Scholar 

  8. 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 

  9. V. E. Zakharov, “A kinetic equation for solitons,” Zh. Eksp. Teor. Fiz.,60, No. 3, 993–1000 (1971).

    Google Scholar 

  10. V. E. Zakharov and S. V. Manakov, “On complete integrability of the KdV equation and the nonlinear Schrödinger equation,” Preprint IYaF, No. 68–73, Novosibirsk (1973).

  11. V. E. Zakharov, L. A. Takhtadzhyan, and L. D. Faddeev, “A complete description of sine-Gordan solutions,” Dokl. Akad. Nauk SSSR,219, No. 6, 1334–1337 (1974).

    Google Scholar 

  12. V. E. Zakharov and L. D. Faddeev, “The Korteweg-de Vries equation — a completely integrable Hamiltonian system,” Funkts. Analiz Prilozhen.,5, No. 4, 18–27 (1971).

    Google Scholar 

  13. V. E. Zakharov and A. B. Shabat, “A scheme for integrating nonlinear equations of mathematical physics by the method of inverse scattering,” Funkts. Analiz Prilozhen.,8, No. 3, 43–53 (1974).

    Google Scholar 

  14. A. P. Its and V. B. Matveev, “On a class of solutions of the Korteweg-de Vries equations,” in: Probl. Mat. Fiz., No. 8, Leningr. Univ., Leningrad (1976), pp. 70–92.

    Google Scholar 

  15. I. M. Krichever, “Potentials with zero reflection coefficients on a background of finite-zone potentials,” Funkts. Analiz Prilozhen.,9, No. 2, 77–78 (1975).

    Google Scholar 

  16. I. M. Krichever, “Integration of nonlinear equations by methods of algebraic geometry,” Funkts. Analiz Prilozhen.,11, No. 1, 15–31 (1977).

    Google Scholar 

  17. E. A. Kuznetsov and A. V. Mikhailov, “On the complete integrability of the classical two-dimensional Thirring model,” Teor. Mat. Fiz.,30, No. 3, 303–314 (1977).

    Google Scholar 

  18. B. A. Kupershmidt, “The Lagrangian formalism in the calculus of variation,” Funkts. Analiz Prilozhen.,10, No. 2, 77–78 (1976).

    Google Scholar 

  19. B. A. Kupershmidt and Yu. I. Manin, “Equations of long waves with a free surface. I. Conservation laws and solutions. II. The Hamiltonian structure and higher equations,” Funkts. Analiz Prilozhen.,11, No. 3, 31–42 (1977).

    Google Scholar 

  20. Yu. I. Manin, “Rational points of algebraic curves over function fields,” Izv. Akad. Nauk SSSR, Ser. Mat.,27, No. 6, 1397–1442 (1963).

    Google Scholar 

  21. S. P. Novikov, “The periodic problem for the Korteweg-de Vries equation. I,” Funkts. Analiz Prilozhen.,8, No. 3, 54–66 (1974).

    Google Scholar 

  22. L. A. Takhtadzhyan and L. D. Faddeev, “An essentially nonlinear one-dimensional model of classical field theory,” Teor. Mat. Fiz.,21, No. 2, 160–174 (1974).

    Google Scholar 

  23. L. D. Faddeev, “The inverse problem of quantum scattering theory. II,” in: Sovr. Probl. Mat. (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1974), pp. 93–180.

    Google Scholar 

  24. L. D. Faddeev, “In search of multidimensional solitons,” in: Nonlocal, Nonlinear, and Nonrenormalizable Field Theories,” JINR (1976), pp. 207–223.

  25. H. Airault, H. P. McKean, and J. J. Moser, “Rational and elliptic solutions of the Korteweg-de Vries equation and a related many-body problem,” Preprint (1976).

  26. R. M. Miura (editor), Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications, NSF Research Workshop on Contact Transformations, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin (1976).

    Google Scholar 

  27. F. Calogero, “Exactly solvable one-dimensional many-body problems,” 1st. di Fisica G. Marconi, Univ. di Roma, Preprint (1975).

  28. F. Calogero and A. Degasperis, “Nonlinear equations solvable by the inverse spectral transform method. II,” Ist. di Fisica G. Marconi, Univ. di Roma, Preprint No. 20 (1976).

  29. F. Galogero and A. Degasperis, “Backlund transformations, nonlinear superposition principles, multisoliton solutions, and conserved quantities for the boomeron nonlinear evolution equation,” Lettere Nuovo Cimento,16, No. 14, 434–438 (1976).

    Google Scholar 

  30. D. V. Choodnovsky and G. V. Choodnovsky, “Pole expansion of nonlinear partial differential equations,” Preprint (1977).

  31. J. Corones, “Solitons and simple pseudopotentials,” J. Math. Phys.,17, No. 5, 1867–1872 (1976).

    Google Scholar 

  32. F. B. Estabrook and H. D. Wahlquist, “Prolongation structures of nonlinear evolution equations,” J. Math. Phys.,17, No. 7, 1293–1297 (1976).

    Google Scholar 

  33. Exact Treatment of Nonlinear Lattice Waves, Progr. Theor. Phys.,59, (1976).

  34. L. D. Faddeev, “Quantization of solitons,” Preprint, Inst. for Adv. Study, Princeton (1976).

    Google Scholar 

  35. L. D. Faddeev, “Some comments on many-dimensional solitons,” Preprint, CERN, TH 2188 (1976).

  36. C. S. Gardner, “The Korteweg-de Vries equation and generalizations. IV. The Korteweg-de Vries equation as a Hamiltonian system,” J. Math. Phys.,12, No. 8, 1548–1551 (1971).

    Google Scholar 

  37. P. Hasenfratz and D. A. Ross, “Are quarks short-range solitons?” Phys. Lett.,64B, No. 1, 78–80 (1976).

    Google Scholar 

  38. R. Hirota, “A direct method of finding exact solutions of nonlinear evolution equations, Lecture Notes in Mathematics, No. 515, Springer-Verlag, Berlin (1976), pp. 40–68.

    Google Scholar 

  39. R. Hirota and J. Satsuma, “A variety of nonlinear equations generated from the Bäcklund transformation for the Toda lattice,” Progr. Theor. Phys., No. 59, 64–100 (1976).

    Google Scholar 

  40. M. D. Kruskal, “The Korteweg-de Vries equation and related evolution equations,” Lect. Appl. Math.,15, 61–83 (1974).

    Google Scholar 

  41. M. D. Kruskal, R. M. Miura, and C. S. Gardner, “Korteweg-de Vries equation and generalizations. V. Uniqueness and nonexistence of polynomial conservation laws,” J. Math. Phys.,11, 952–960 (1970).

    Google Scholar 

  42. P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math.,21, No. 5, 467–490 (1968).

    Google Scholar 

  43. P. D. Lax, “Periodic solutions of the KdV equations,” Comm. Pure Appl. Math.,28, No. 1, 141–188 (1975).

    Google Scholar 

  44. V. B. Matveev, “Abelian functions and solitons,” Inst. Theor. Phys., Univ. Wroclaw, Preprint No. 373 (1976).

  45. H. P. McKean and E. Trubowitz, “Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points,” Comm. Pure Appl. Math.,29, No. 2, 143–226 (1976).

    Google Scholar 

  46. R. M. Miura, “Conservation laws for the fully nonlinear long wave equations,” Stud. Appl. Math.,53, No. 1, 45–56 (1974).

    Google Scholar 

  47. R. Miura, “The Korteweg-de Vries equation: a survey of results,” SIAM Rev.,18, No. 3, 412–459 (1976).

    Google Scholar 

  48. R. M. Miura, C. S. Gardner, and M. D. Kruskal, “Korteweg-de Vries equations and generalizations. II. Existence of conservation laws and constants of the motion,” J. Math. Phys.,9, No. 8, 1204–1209 (1968).

    Google Scholar 

  49. H. C. Morris, “Prolongation structures and a generalized inverse scattering problem,” J. Math. Phys.,17, No. 10, 1867–1869 (1976).

    Google Scholar 

  50. Nonlinear Waves, Lect. Appl. Math. 1974, AMS, Providence (1974).

  51. A. Patani, M. Schlinwein, and Q. Shafi, “Topological charges in the field theory,” J. Phys. A: Math. and Gen.,9, 1513–1520 (1976).

    Google Scholar 

  52. K. Pohlmeyer, “Integrable Hamiltonian systems and interaction through quadratic constraints,” Comm. Math. Phys.,46, 207–221 (1976).

    Google Scholar 

  53. A. C. Scott, F. Y. F. Chu, and D. W. MacLaughlin, “The soliton: a new concept in applied science,” Proc. IEEE,61, 1443–1483 (1973).

    Google Scholar 

  54. H. D. Wahlquist and F. B. Estabrook, Prolongation structures of nonlinear evolution equations. I,” J. Math. Phys.,16, No. 1, 1–7 (1975).

    Google Scholar 

Download references

Authors

Additional information

Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 11, pp. 5–152, 1978.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Manin, Y.I. Algebraic aspects of nonlinear differential equations. J Math Sci 11, 1–122 (1979). https://doi.org/10.1007/BF01084246

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01084246

Keywords

Navigation