Geometry of the modulational instability. III: Homoclinic orbits for the periodic sine-Gordon equation. (English) Zbl 0705.58026

Summary: [See also the authors, part I: Local analysis, part II: Global analysis. Univ. Arizona, Preprint (1987).]
The homoclinic geometric structure of the integrable sine-Gordon equation under periodic boundary conditions is developed. Specifically, focus is given to orbits homoclinic to N-tori. Simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given. A labeling is provided which identifies and catalogues all such orbits. These orbits are related in a one-to-one manner to linearized instabilities. Explicit formulas for all homoclinic orbits are given in terms of Bäcklund transformations.


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.)
37C75 Stability theory for smooth dynamical systems
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
37A30 Ergodic theorems, spectral theory, Markov operators
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI


[1] Ablowitz, M. J.; Kaup, D.; Newell, A.; Segur, H., Phys. Rev. Lett., 30, 1262 (1973)
[2] Baker, H. F., (Proc. R. Soc. London Ser. A, 118 (1928)), 584-593
[3] Benjamin, T. B., The disintegration of wavetrains on deep water, J. Fluid Mech., 27, 417 (1966) · Zbl 0144.47101
[4] Benjamin, T. B., Instabilities of periodic wave trains in nonlinear dispersive systems, (Proc. R. Soc. London Ser. A, 299 (1967)), 59-75
[5] Benney, D. J.; Newell, A., Propagation of nonlinear wave envelopes, J. Math. Phys., 46, 133-139 (1967) · Zbl 0153.30301
[6] Bishop, A. R.; Fesser, K.; Lomdahl, P. S.; Kerr, W. C.; Williams, M. B.; Tullinger, S. E., Coherent spatial structure versus time chaos in a perturbed sine-Gordon system, Phys. Rev. Lett., 50, 1095-1099 (1983)
[7] Bishop, A. R.; Fesser, K.; Lomdahl, P. S.; Trullinger, S. E., Influence of solitons in the initial state on chaos and the driven, damped sine-Gordon system, Physica D, 7, 259-279 (1983) · Zbl 1194.35344
[8] Bishop, A. R.; Forest, M. G.; McLaughlin, D. W.; Overman, E. A., Quasi-periodic route to chaos in a near integrable PDE, Physica D, 23, 293-328 (1986) · Zbl 0616.65135
[9] Bishop, A. R.; Forest, M. G.; McLaughlin, D. W.; Overman, E. A., A quasi-periodic route to chaos in a near-integrable PDE: homoclinic crossings, Phys. Lett. A, 127, 335-340 (1988)
[10] Bishop, A. R.; Grüner, G.; Nicolaenko, B., Spatio-temporal coherence and chaos in physical systems, Physica D, 23 (1986), special issue · Zbl 0604.00013
[11] (Campbell, D.; Rose, H., Order in chaos. Order in chaos, Physica D, 7 (1983)), special issue · Zbl 0536.00007
[12] Chen, H. H.; Lee, Y. C.; Tracy, E., Study of quasiperiodic solutions of the nonlinear Schrödinger equation and the nonlinear modulational instability, Phys. Rev. Lett., 53, 218-221 (1984)
[13] Date, E., Multisoliton solutions and quasi-periodic solutions of nonlinear equations of sine-Gordon type, Osaka J. Math., 19, 125-128 (1982) · Zbl 0505.35076
[14] Doolen, G. D.; Dubois, D. F.; Rose, H., Coherence and chaos in caviton turbulence, Phys. Rev. Lett., 51, 335 (1983)
[15] Ercolani, N. M.; Flaschka, H., The geometry of the Hill equation and of the Neumann system, Phil. Trans. R. Soc. London Ser. A, 315, 405-422 (1985) · Zbl 0582.35105
[16] Ercolani, N. M.; Forest, M. G., The geometry of real sine-Gordon wavetrains, Comm. Math. Phys., 99, 1-45 (1985) · Zbl 0591.35065
[17] Ercolani, N. M.; Forest, M. G.; McLaughlin, D. W., Modulational instabilities of periodic sine-Gordon waves: a geometric analysis, Lect. Appl. Math., 23, 149-165 (1986) · Zbl 0597.58011
[18] Ercolani, N. M.; Forest, M. G.; McLaughlin, D. W., The origin and saturation of modulational instabilities, Physica D, 18, 472-474 (1986) · Zbl 0596.35112
[19] Ercolani, N. M.; Forest, M. G.; McLaughlin, D. W., Geometry of the modulational instability I. Local analysis, University of Arizona Preprint (1987) · Zbl 0597.58011
[20] Ercolani, N. M.; Forest, M. G.; McLaughlin, D. W., Geometry of the modulational instability II. Global analysis, University of Arizona Preprint (1987) · Zbl 0597.58011
[21] Finkel, A., Remarks on an eigenvalue problem associated to the sine-Gordon equation, (Thesis (1982), New York University)
[22] Flaschka, H.; McLaughlin, D. W., Some comments on Bäcklund transformations and inverse scattering method, (Miura, R., Proceedings of the NSF Workshop on Contract Transformations. Proceedings of the NSF Workshop on Contract Transformations, Springer Lecture Notes, No. 515 (1975), Springer: Springer Berlin) · Zbl 0346.35027
[23] Forest, M. G.; McLaughlin, D. W., Spectral theory for the periodic sine-Gordon equation: a concrete viewpoint, J. Math. Phys., 23, 1248-1277 (1982) · Zbl 0498.35072
[24] Forest, M. G.; McLaughlin, D. W., Modulation of sine-Gordon and sinh-Gordon wavetrains, Stud. Appl. Math., 68, 11-59 (1983) · Zbl 0541.35071
[25] Guckenheimer, J.; Holmes, P., Nonlinear Oscillations, Dynamical System, and Bifurcations of Vector Fields (1983), Springer: Springer Berlin · Zbl 0515.34001
[27] McKean, H. P., Stability for the Korteweg-de Vries equation, Comm. Pure Appl. Math., 30, 347-353 (1977) · Zbl 0335.58013
[28] McKean, H. P., The sine-Gordon and sinh-Gordon equations on the circle, Comm. Pure Appl. Math., 34, 197-257 (1981) · Zbl 0467.35078
[29] McKean, H. P.; Trubowitz, E., Hill’s operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math., 29, 143-226 (1976) · Zbl 0339.34024
[30] Moon, H. T.; Goldman, M. V., Intermittency and solitons in the driven dissipative NLS equation, Phys. Rev. Lett., 53, 1821 (1984)
[31] Moon, H. T.; Huerre, P.; Redekopp, L. G., Transitions to chaos in the Ginzburg-Landau equation, Physica D, 7, 135 (1983) · Zbl 0558.58030
[33] Overman, E. A.; McLaughlin, D. W.; Bishop, A. R., Coherence and chaos in the driven, damped sine-Gordon equation: measurement of soliton spectrum, Physica D, 19, 1-41 (1986) · Zbl 0615.35071
[34] Takhtajian, L. A.; Faddeev, L. D., Essentially nonlinear one-dimensional model of classical field theory, Theor. Math. Phys., 21, 1046 (1974) · Zbl 0299.35063
[35] Wadati, M.; Sanuki, H.; Konno, K., Relationships among inverse method, Bäcklund transformation, and an infinite number of conservation laws, Prog. Theor. Phys., 53, 419-436 (1975) · Zbl 1079.35506
[36] Whitham, G. B., Linear and Nonlinear Waves (1974), Wiley-Interscience: Wiley-Interscience New York · Zbl 0373.76001
[37] Whittaker, E. T.; Watson, G. N., Modern Analysis (1902), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0108.26903
[38] Zakharov, V. E., Ph.D. Thesis, ((1966), Institute of Nuclear Physics, Siberian Division, USSR Academy of Science)
[39] Akhmediev, N. N.; Eleonshii, V. M.; Kulagin, N. E., Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions, Zh. Eksp. Teor. Fiz., 89, 1542-1551 (1985)
[40] Akhmediev, N. N.; Korneev, V. I.; Mitskevich, N. V., \(N\)-modulation signals in a single-mode optical waveguide under nonlinear conditions, Zh. Eksp. Teor. Fiz., 94, 159-170 (1988)
[41] Bishop, A. R.; Forest, M. G.; McLaughlin, D. W.; Overman, E. A., A modal representation of chaotic attractors for the driven damped pendulum chain, Phys. Lett. A, 144, 335-340 (1988)
[42] Driscoll, C.; O’Neill, T., Explanations of instabilities on a Fermi-Pasta-Ulam lattice, Phys. Rev. Lett., 37, 69-72 (1976)
[43] Flesch, R.; Forest, M. G.; Sinha, A., Numerical IST for the periodic sine-Gordon equation: theta function solutions and their linearized stability (1990), Ohio State University, Preprint
[44] Kovacic, G.; Wiggins, S., Orbits homoclinic to resonances: chaos in a model of the forced and damped sine-Gordon equation (1989), California Institute of Technology, Preprint
[45] Moon, H. T., Homoclinic crossings and pattern selection, Phys. Rev. Lett., 64, 412-414 (1990)
[46] Tracy, E.; Chen, H. H., Nonlinear self-modulation: an exact solvale model, Phys. Rev. A, 37, 815 (1988)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.