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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M.J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia 1981).
M. Adler and P. Van Moerbeke, Completely integrable systems, Kac-Moody Lie algebras and curves, Adv.in Math. 38(3): 267(1980).
M.S. Alber and J.E.Marsden, On Geometric Phases for Soliton Equations, Commun. Math. Phys. (to appear).
M.S. Alber and J.E.Marsden, Umbilic Solitons and Homoclinic Geometric Phases, (in print).
M.S. Alber, On integrable systems and semiclassical solutions of the stationary Schrödinger equations, Inverse Problems 5: 131 (1989).
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).
M.S. Alber, Hyperbolic Geometric Asymptotics, Asymptotic Analysis 5, 2: 161 (1991).
M.S. Alber and S.J. Alber, Hamiltonian formalism for finite-zone solutions of integrable equations, C.R. Acad. Sc. Paris 301: 777 (1985).
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).
V.I. Arnold, A remark on the branching of hyperelliptic integrals as functions of the parameters, Func. Anal. Appl. 2: 187 (1968).
M.V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond.A 392: 45 (1984).
M.V. Berry, Classical adiabatic angles and quantal adiabatic phase, J. Phys. A: Math. Gen. 18: 15 (1985).
M.V. Berry and J.H. Hannay, Classical non-adiabatic angles, J.Phys.A: Math. Gen. 21: L325 (1988).
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.
B.Birnir, Singularities of the complex Korteweg-de Vries flows, Comm. Pure Appl. Math. 39: 283 (1986).
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).
P. Deift, L.C. Li and C. Tomei, Matrix factorizations and integrable systems, Comm. Pure Appl. Math. 443 (1989).
R. Devaney, Transversal homoclinic orbits in an integrable system, Amer. J. Math., 100: 631 (1978).
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).
S.U.Dobrohotov and V.P.Maslov, Finite-zone, almost-periodic solutions in WKB approximation, J.Soviet Math.,16, 6: 1433 (1981).
H.J. Duistermaat, On global Action-Angle Coordinates, Comm. Pure Appl. Math. 23: 687 (1980).
N. Ercolani, Generalized Theta functions and homoclinic varieties, Proc. Symp. Pure Appt. Math., 49: 87 (1989).
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).
N. Ercolani and H. Flaschka, The geometry of the Hill equation and of the Neumann system, Phil. Trans. R. Lond. A 315:405 (1985).
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).
N. Ercolani, M. Forest and D.W. McLaughlin, Notes on Melnikov integrals for models of the periodic driven pendulum chain, (preprint) (1989).
N. Ercolani and M. Forest, The Geometry of Real Sine-Gordon Wavetrains, Comm. Math. Phys. 99(1985)1–49.
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).
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).
H. Flaschka, A.C. Newell and T. Ratiu, Kac-Moody Lie algebras and soliton equations II, III, Physica D 9: 300 (1983).
M.G. Forest and D.W. McLaughlin, Modulation of Sinh-Gordon and Sine-Gordon Wavetrains, Studies in Appl. Math. 68: 11 (1983).
V. Guillemin and S. Sternberg, Geometric Asymptotics, Math.Sutveys 14, AMS, Providence, Rhode Island, (1977).
V. Guillemin and S. Sternberg, The Gelfand-Cetlin system and the quantization of the complex Flag Manifolds, J. Funct. Anal. 52 (1): 106 (1983).
W.Klingenberg, Riemannian Geometry, Berlin: New York: de Gruyter (1982).
H. Knörrer, Singular fibres of the momentum mapping for integrable Hamiltonian systems, J. Reine u. Ang. Math. 355: 67 (1984).
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).
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).
H.P. McKean, Integrable Systems and Algebraic Curves, Lecture Notes in Mathematics, Springer-Verlag, Berlin, (1979).
H.P. McKean, Theta functions, solitons, and singular curves, in, PDE and Geometry, Proc. of Park City Conference, C.I.Byrnes, ed., (1977).
D.W. McLaughlin and E.A. Overman, Surveys in Appl.Math. 1 (1992) (to appear).
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).
J. Moser, Integrable Hamiltonian Systems and Spectral Theory, Academia Nazionale dei Lincei, Pisa, (1981).
D. Mumford, Tata Lectures on Theta I and II, Progress in Math.28 and 43, Birkhauser, Boston (1983).
E. Previato, Hyperelliptic quasi-periodic and soliton solutions of the nonlinear Schrödinger equation, Duke Math.Journal 52: 2 (1985).
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).
S. Venakides, The generation of modulated wavetrains in the solution of the Korteweg-de Vries equation, Comm.Pure and Appl.Math. 38: 883 (1985).
S. Venakides and T. Zhang, Periodic limit of inverse scattering, Duke University, preprint (1991).
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).
A. Weinstein, Connections of Berry and Hannay type for moving Lagrangian sub-manifolds, Adv. in Math. 82, 2: 133 (1990).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights 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