Summary
We describe examples of two phenomena: the decay of a nonlinear coherent state under the influence of friction and the resonance of such a state due to a time periodic external forcing. In the latter case a period doubling transition to chaotic motion can be found as function of a parameter in the forcing. Essential is that in each case the state remains spatially coherent. The main features of these phenomena can be described with only two observables: say the amplitude and the phase of the coherent state. Here one method is given to choose these observables properly and to formulate their equations of motion in each case.
This is a preview of subscription content, log in via an institution to check access.
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
W.E. Ferguson Jr.; H. Flaschka; D.W. McLaughlin: Nonlinear normal modes for the Toda chain. J. Comp. Phys. 45 (1982) 157–209.
T.P. Valkering: Periodic travelling waves in a non-integrable one-dimensional lattice. Cel. Mech. 28 (1982) 119–131.
G.B. Whitham: Linear and nonlinear waves. New York: Wiley 1974
G.J. Brinkman;T.P. Valkering: Soliton decay in a Toda chain caused by dissipation. To appear in J. Appl. Math. Phys. ZAMP.
E. van Groesen; F van Beckum; T.P. Valkering: Decay of travelling waves in dissipative Poisson systems. Submitted (1989).
J.H.J. van Opheusden; T.P. Valkering: Period doubling density waves in a chain. Nonlinearity 2 (1989) 357–371.
T.P. Valkering; G. Derks; E. van Groesen: Modelling one degree of freedom behaviour of dissipative Hamiltonian systems with forcing. In: St. Pneumatikos; T. Bountis; Sp. Pneumatikos (eds.): Proc of the. Intern. Conf. on Singular Behaviour and Nonlinear Dynamics, Samos, Greece. World Sc Publ. Corp. 1989.
A.J. Lichtenberg; M.A. Lieberman: Regular and Stochastic Motion. New York: Springer 1983.
N.B. Tuffilaro; A.M. Albano: Chaotic dynamics of a bouncing ball. Am. J. Phys. 54 (1986) 939–944.
K. Nozaki; N. Bekki: Low-dimensional chaos in a driven damped nonlinear Schrödinger equation. Physica 21D (1986) 381–393.
M. Taki; K.H. Spatchek; J.C. Fernandez; R. Grauer; G. Reinisch: Breather dynamics in the nonlinear Schrödinger regime of perturbed Sine-Gordon systems. Physica D (1989) in press.
P.D. Lax: Integrals of nonlinear quations of evolution and solitary waves. Comm. Pure Appl. Math. 21 (1968) 467–490.
K. Geist; W. Lauterborn: The nonlinear dynamics of the damped and driven Toda chain. Physica D 31 (1988) 103–116.
Th. Zeegers; T.P. Valkering: The period doubling sequence of (semi) symplectic oscillators coupled to a linear one. In preparation.
D.K. Campbell; J.F. Schonfeld; C.A. Wingate: Resonance structure in kinkanti-kink interactions in (I)4 theory. Physica 9D (1983) 1–32.
P.R. Sethna: On symmetry breaking bifurcations: Local and global phenomena. These Proceedings.
J. Cusunamo; F.C. Moon: Low-dimensional behaviour in chaotic nonplanar motions of a forced linearly elstic rod: Experiment and theory. These Proceedings.
S. Ciliberto; J.P. Gollub. Chaotic mode competition in parametrically forced surface waves. J. Fluid Mech. (1985) 381–398.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1990 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Valkering, T.P., van Groesen, E. (1990). Decay and Resonance of Coherent States. In: Schiehlen, W. (eds) Nonlinear Dynamics in Engineering Systems. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-83578-0_41
Download citation
DOI: https://doi.org/10.1007/978-3-642-83578-0_41
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-83580-3
Online ISBN: 978-3-642-83578-0
eBook Packages: Springer Book Archive