Skip to main content

Geometric Phases and Monodromy at Singularities

  • Chapter
Book cover Singular Limits of Dispersive Waves

Part of the book series: NATO ASI Series ((NSSB,volume 320))

Abstract

In [14, 25, 22, 28, 40] and [3, 4] methods of the complex analysis and algebraic geometry were applied for investigating the geometry of the phase spaces of the nonlinear systems.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia 1981).

    Book  MATH  Google Scholar 

  2. M. Adler and P. Van Moerbeke, Completely integrable systems, Kac-Moody Lie algebras and curves, Adv.in Math. 38(3): 267(1980).

    Article  MathSciNet  MATH  Google Scholar 

  3. M.S. Alber and J.E.Marsden, On Geometric Phases for Soliton Equations, Commun. Math. Phys. (to appear).

    Google Scholar 

  4. M.S. Alber and J.E.Marsden, Umbilic Solitons and Homoclinic Geometric Phases, (in print).

    Google Scholar 

  5. M.S. Alber, On integrable systems and semiclassical solutions of the stationary Schrödinger equations, Inverse Problems 5: 131 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  6. M.S. Alber, Complex Geometric Asymptotics, Geometric Phases and Nonlinear Integrable Systems, in, Studies in Math.Phys. 3, North-Holland, Elsevier Science Publishers B.V., Amsterdam (1992).

    Google Scholar 

  7. M.S. Alber, Hyperbolic Geometric Asymptotics, Asymptotic Analysis 5, 2: 161 (1991).

    MathSciNet  MATH  Google Scholar 

  8. M.S. Alber and S.J. Alber, Hamiltonian formalism for finite-zone solutions of integrable equations, C.R. Acad. Sc. Paris 301: 777 (1985).

    MathSciNet  Google Scholar 

  9. M.S. Alber and S.J. Alber, Hamiltonian formalism for nonlinear Schrödinger equations and sine-Gordon equations, J.London Math.Soc. (2) 36: 176 (1987).

    Article  MathSciNet  Google Scholar 

  10. V.I. Arnold, A remark on the branching of hyperelliptic integrals as functions of the parameters, Func. Anal. Appl. 2: 187 (1968).

    Article  Google Scholar 

  11. M.V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond.A 392: 45 (1984).

    Article  ADS  MATH  Google Scholar 

  12. M.V. Berry, Classical adiabatic angles and quantal adiabatic phase, J. Phys. A: Math. Gen. 18: 15 (1985).

    Article  ADS  MATH  Google Scholar 

  13. M.V. Berry and J.H. Hannay, Classical non-adiabatic angles, J.Phys.A: Math. Gen. 21: L325 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. M.V.Berry, Quantum Adiabatic Anholonomy, in, Lectures given at the Ferrara School of Theoretical Physics on “Anomalies, defects, phases…”, June 1989, (to be published by Bibliopolis), Naples.

    Google Scholar 

  15. B.Birnir, Singularities of the complex Korteweg-de Vries flows, Comm. Pure Appl. Math. 39: 283 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  16. A.R. Bishop, D.W. McLaughlin and M. Solerno, Global coordinates for the breather-kink (antikink) sine-Gordon phase space: An explicit separatrix as a possible source of chaos, Phys. Rev. A, 40, 11: 6463 (1989).

    Article  MathSciNet  ADS  Google Scholar 

  17. P. Deift, L.C. Li and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math. 443 (1989).

    Google Scholar 

  18. R. Devaney, Transversal homoclinic orbits in an integrable system, Amer. J. Math., 100: 631 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  19. S.U.Dobrohotov and V.P.Maslov, Multiphase asymptotics of nonlinear partial differential equations with a small parameter, in, Math. Phys. Reviews, Over. Pub. Ass., Amsterdam, (1982).

    Google Scholar 

  20. S.U.Dobrohotov and V.P.Maslov, Finite-zone, almost-periodic solutions in WKB approximation, J.Soviet Math.,16, 6: 1433 (1981).

    Article  Google Scholar 

  21. H.J. Duistermaat, On global Action-Angle Coordinates, Comm. Pure Appl. Math. 23: 687 (1980).

    Article  MathSciNet  Google Scholar 

  22. N. Ercolani, Generalized Theta functions and homoclinic varieties, Proc. Symp. Pure Appt. Math., 49: 87 (1989).

    Article  MathSciNet  Google Scholar 

  23. N. Ercolani and H.P. McKean, Geometry of KdV(4): Abel sums, Jacobi variety, and theta function in the scattering case, Invent. Math. 99: 483 (1990).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. N. Ercolani and H. Flaschka, The geometry of the Hill equation and of the Neumann system, Phil. Trans. R. Lond. A 315:405 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  25. N. Ercolani, M. Forest, D.W. McLaughlin, and R. Montgomery, Hamiltonian structure for the modulation equations of a sine-Gordon wavetrain, Duke Math. Journal 55, 4:949 (1987).

    Article  MATH  Google Scholar 

  26. N. Ercolani, M. Forest and D.W. McLaughlin, Notes on Melnikov integrals for models of the periodic driven pendulum chain, (preprint) (1989).

    Google Scholar 

  27. N. Ercolani and M. Forest, The Geometry of Real Sine-Gordon Wavetrains, Comm. Math. Phys. 99(1985)1–49.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. N. Ercolani and D.W. McLaughlin, Toward a topological classification of integrable PDE’s, in, Proc. of a Workshop on The Geometry of Hamiltonian Systems, T.Ratiu, ed., MSRI Publications 22, Springer-Verlag, (1991).

    Google Scholar 

  29. H. Flaschka, D.W. McLaughlin and M.G. Forest, Multiphase averaging and the inverse spectral solution of the Korteweg-de Vries equation, Comm. Pure Appl. Math. 33: 739 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. H. Flaschka, A.C. Newell and T. Ratiu, Kac-Moody Lie algebras and soliton equations II, III, Physica D 9: 300 (1983).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. M.G. Forest and D.W. McLaughlin, Modulation of Sinh-Gordon and Sine-Gordon Wavetrains, Studies in Appl. Math. 68: 11 (1983).

    MATH  Google Scholar 

  32. V. Guillemin and S. Sternberg, Geometric Asymptotics, Math.Sutveys 14, AMS, Providence, Rhode Island, (1977).

    Google Scholar 

  33. V. Guillemin and S. Sternberg, The Gelfand-Cetlin system and the quantization of the complex Flag Manifolds, J. Funct. Anal. 52 (1): 106 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  34. W.Klingenberg, Riemannian Geometry, Berlin: New York: de Gruyter (1982).

    MATH  Google Scholar 

  35. H. Knörrer, Singular fibres of the momentum mapping for integrable Hamiltonian systems, J. Reine u. Ang. Math. 355: 67 (1984).

    Google Scholar 

  36. P.D. Lax and C.D. Levermore, The small Dispersion Limit of the Korteweg - de Vries Equation, I, II, III, Comm. Pure Appl. Math. 36, 2: 253, 571, 809 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  37. J.E. Marsden, R. Montgomery and T. Ratiu, Cartan-Hannay-Berry phases and symmetry, Contemporary Mathematics 97: 279 (1989); see also Mem. AMS Vol. 436 (1990).

    Article  MathSciNet  Google Scholar 

  38. H.P. McKean, Integrable Systems and Algebraic Curves, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (1979).

    Google Scholar 

  39. H.P. McKean, Theta functions, solitons, and singular curves, in, PDE and Geometry, Proc. of Park City Conference, C.I.Byrnes, ed., (1977).

    Google Scholar 

  40. D.W. McLaughlin and E.A. Overman, Surveys in Appl.Math. 1 (1992) (to appear).

    Google Scholar 

  41. R. Montgomery, The connection whose holonomy is the classical adiabatic angles of Hannay and Berry and its generalization to the non-integrable case, Comm. Math. Phys. 120: 269 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  42. J. Moser, Integrable Hamiltonian Systems and Spectral Theory, Academia Nazionale dei Lincei, Pisa, (1981).

    MATH  Google Scholar 

  43. D. Mumford, Tata Lectures on Theta I and II, Progress in Math.28 and 43, Birkhauser, Boston (1983).

    Google Scholar 

  44. E. Previato, Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math.Journal 52: 2 (1985).

    Article  MathSciNet  Google Scholar 

  45. S.N.M. Ruijsenaars, Action-angle maps and scattering theory for some finite-dimensional integrable systems I.The pure soliton case, Comm. Math. Phys. 115: 127 (1988).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  46. S. Venakides, The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation, Comm.Pure and Appl.Math. 38: 883 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  47. S. Venakides and T. Zhang, Periodic limit of inverse scattering, Duke University, preprint (1991).

    Google Scholar 

  48. Yu.M.Vorob’ev and S.U.Dobrohotov, Quasiclassical quantisation of the periodic Toda chain from the point of view of Lie algebras, Theor.and Math.Phys. 54, 3:312 (1983).

    Article  ADS  MATH  Google Scholar 

  49. A. Weinstein, Connections of Berry and Hannay type for moving Lagrangian sub-manifolds, Adv. in Math. 82, 2: 133 (1990).

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer Science+Business Media New York

About this chapter

Cite this chapter

Alber, M.S., Marsden, J.E. (1994). Geometric Phases and Monodromy at Singularities. In: Ercolani, N.M., Gabitov, I.R., Levermore, C.D., Serre, D. (eds) Singular Limits of Dispersive Waves. NATO ASI Series, vol 320. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-2474-8_20

Download citation

  • DOI: https://doi.org/10.1007/978-1-4615-2474-8_20

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-6054-4

  • Online ISBN: 978-1-4615-2474-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics