Abstract
Previous numerical studies of the damped, driven sine-Gordon equation with spatially periodic boundary conditions have identified various low-dimensional attractors and bifurcation phenomena. These attractors are fully nonlinear (i.e., order one amplitude) space-time structures which have been independently measured (using the spectral transform) in terms of integrable, sine-Gordon modes. Based on these direct measurements, we posit a leading order approximation to the perturbed flow in terms of modulated sine-Gordon wavetrains. Our goal here is to present dynamical simulations of two-phase perturbed sine-Gordon modulation equations and to compare these predictions with the results of direct pde simulations using Ed Overman’s codes.
1 Research supported by NSF DMS 88-03465, 91-04806.
2 We acknowledge computer support on the CRAY YMP from the Ohio Supercomputer Center.
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References
A. R. Bishop, M. G. Forest, D. W. McLaughlin and E. A. Overman II, A quasiperiodic route to chaos in a near-integrable pde, Physica D 23:293(1986); A quasiperiodic route to chaos in a near-integrable pde: homoclinic crossings, Physics Letters A 127:335 (1988).
A. R. Bishop, D. W. McLaughlin and E. A. Overman II, Coherence and chaos in the driven damped sine-Gordon equation: measurement of soliton spectrum, Physica D 19:1 (1986).
A. R. Bishop, R. Flesch, M. G. Forest, D. W. McLaughlin and E. A. Overman II, Correlations between chaos in a perturbed sine-Gordon equation and a truncated modal system, SIAM J. Math. Anal. 23:1511 (1990); C. Xiong, Ph. D. Thesis, The Ohio State University (1991).
[4 N. M. Ercolani, M. G. Forest, D. W. McLaughlin and A. Sinha, Fully nonlinear modal equations for nearly integrable PDEs,Journal of Nonlinear Science, to appear.
[5 R. Flesch, M. G. Forest and A. Sinha, Numerical inverse spectral transform for the periodic sine-Gordon equation: theta function solutions and their linearized stability, Physica D 48:169 (1991).
[6 N. M. Ercolani and M. G. Forest, The geometry of real sine-Gordon wavetrains, Comm. Math. Phys. 99:1 (1985).
[7 M. G. Forest, S.-P. Sheu and A. Sinha, Frequency and phase locking of spatially periodic perturbed sine-Gordon breather trains, SIAM J. on Appl. Math., 52:746 (1992).
E. A. Overman, Private communication.
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© 1994 Springer Science+Business Media New York
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Forest, M.G., Sinha, A. (1994). A Numerical Study of Nearly Integrable Modulation Equations. 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_3
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DOI: https://doi.org/10.1007/978-1-4615-2474-8_3
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