Abstract
We consider spatially inhomogeneous Hamiltonian systems for which the rate of change of the inhomogeneity is small. Connected to these systems is a 1-parameter family of homogenized versions, for which spatial variations vanish. Special solutions of these homogenized systems are relative equilibrium solutions: a 2-parameter manifold of solutions which are translations of an extremizer of the energy constrained to levelsets of momentum. A solution of the inhomogeneous system which describes the distortion of such a relative equilibrium solution is approximated using relative equilibrium states with the 3 parameters evolving in time in a way to be specified. The dynamics of the parameters is obtained using (i) a geometrically motivated projection argument, (ii) a dynamical consistent evolution of global quantities (energy and momentum), and (iii) a Fredholm-type of argument from a mathematical investigation of the error. The results are shown to be equivalent. The Fredholm-argument implies that the approximation is valid on spatial-temporal scales on which deformations are of order one, thereby justifying the physically more attractive method of consistent evolution. All results are illustrated to the motion of a Bloch wall in an inhomogeneous ferro-magnetic material.
This research has been supported by the Netherlands Organization for Scientific Research, NWO, by contract 620-61-249.
Part of the research is sponsored by the Commission of the European Communities, Directorate General XII-B, Joint Research Project CI1*-CT93-0018 between the Department of Mathematics. Institut Teknologi Bandung, Indonesia, and the Faculty of Applied Mathematics, University of Twente, The Netherlands.
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References
Aceves, A., Adachihara, H., Jones, C., Lerman, J.C., McLaughlin, D.W., Moloney, J.V., and Newell, A.C. Chaos and coherent structures in partial differential equations. Physica 18(D), 85–112, 1986.
Blommers, B., and Booij, W. Periodic solition solutions for a perturbed sine-Gordon equation (in Dutch). Technical report, University of Twente, 1992.
Buitelaar, R.P. The method of averaging in Banach spaces: theory and application. PhD-thesis, Univ. of Utrecht, 1993.
Calogero, F., and Eckhaus, W. Nonlinear evolution equations, rescalings, model PDE’s and their integrability: II. Inverse Problems, 4, 11–33, 1988.
Carr, J. Applications of Centre Manifold Theory. Springer-Verlag, New York, 1981.
Derks, G. Coherent structures in the dynamics of perturbed Hamiltonian systems. PhD-thesis, University of Twente, 1992.
Derks, G., and Groesen, E. van. Dissipation in Hamiltonian systems: Decaying cnoidal waves. To be pubished, SIAM J. Math. Anal., 1995.
Fledderus, E.R., and Groesen, E. van. Critical swirling flows in expanding pipes, Part II. To be published, SIAM J. Math. An. Appl., 1995.
Fledderus, E.R., and Groesen, E. van. Deformation of coherent structures in inho-mogeneous media. Rep. Progr. Phys., forthcoming, 1995.
Groesen, E. van, Beckum, F.P.H. van, and Valkering, T.P. Decay of travelling waves in dissipative Poisson systems. ZAMP, 41, 501–523, 1990.
Groesen, E. van, Fliert, B.W. van de, and Fledderus, E. Quasi-homogeneous critical swirling flows in expanding pipes, I. SIAM J. Math. An. Appl., 192. 764–788, 1995.
Groesen, E. van. A Hamiltonian Perturbation Theory for Coherent Structures illustrated to wave problems. Proceedings of the IUTAM/ISIMM Symposium, Hannover, Germany, 1995.
Leeuw F.H. de, Doei, R. van den, and Enz, U. Dynamic properties of magnetic domain walls and magnetic bubbles. Rep. Prog. Phys. 43, 689–783. 1980.
Hale, J.K. Ordinary differential equations. Wiley-Interscience, New York, 1969.
Karpman, V.I., and Solov’ev, V.V. The influence of perturbations on the shape of a sine-Gordon soliton. Phys. Lett., 84A(2), 39–41, 1981.
Kato, T. Perturbation Theory for Linear Operators. Springer-Verlag, New York, 1966.
Kivshar, Y.S., and Malomed, B.A. Dynamics of solitons in nearly integrable systems. Rev. Mod. Phys. 61(4), 763–915, 1989.
Krol, M.S. The Method of Averaging in Partial Differential Equations. PhD-thesis, Univ. of Utrecht, 1990.
Nayfeh, A.H. Introduction to Perturbation Techniques. Wiley, New York, 1993.
Olsen, O.H., and Samuelsen, M.R. Sine-Gordon 27r-kink dynamics in the presence of small perturbations. Phys. Rev. B, 28(1), 210–217, 1983.
Pudjaprasetya, S.R., and Groesen, E. van. Uni-directional waves over slowly varying bottom. Part II: Quasi-homogeneous approximation of distorting waves. To be published, Wave Motion, 1995.
Sakai, S., Samuelsen, M.R., and Olsen, O.H. Perturbation analysis of a parametrically changed sine-Gordon equation. Phys. Rev. B, 36(1), 217–225, 1987.
Salerno, M., Samuelsen, M.R., Lomdahl, P.S., and Olsen O.H. Non-dissipative perturbations in the sine-Gordon system. Phys. Lett. 108A, 241–244, 1985.
Sanders, J.A., and Verludst, F. Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York, 1985.
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Fledderus, E.R., van Groesen, E. (1996). Hamiltonian Perturbation Theory for Concentrated Structures in Inhomogeneous Media. In: Broer, H.W., van Gils, S.A., Hoveijn, I., Takens, F. (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7518-9_16
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DOI: https://doi.org/10.1007/978-3-0348-7518-9_16
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