Abstract
Elastic crystals with a microstructure include ferroelectric crystals, ferromagnetic crystals and crystals with internal mechanical degrees of freedom. In recent works concerning the discrete or continuum modelling of the behavior of such elastic crystals, we have been able to delineate ā general descriptive framework in which,using the concepts of solitary waves and solitons, the dynamics of simple structures in domains and walls can be accommodated. To that purpose the notion of Bloch and Néel walls in deformable crystals was introduced in all cases. The general nonlinear mathematical problem obtained concerns a nonlinear hyperbolic dispersive system made of a sine-Gordon equation, or a double sine-Gordon equation, for an internal parameter related to the microstructure, which is nonlinearly coupled to one or two wave equations governing elastic displacements. While exact stable nonlinear solutions of the solitary-wave type can be exhibited in a more or less straightforward manner the problem of the interaction of such wave motions (representing then the collision of walls and the coalescence of domains) and that of the transient motion of such waves when acted upon by an external stimulus (then representing the starting motion of walls) can be tackled only by using more sophisticated methods such as singular perturbations, Whitham's averaged Lagrangian method and those methods familiar in soliton theory (Blacklund transformations, inverse scattering method). This is the concern of the present lecture with an emphasis on perturbations.
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References
J. Pouget and G. A. Maugin, Solitons and Electroacoustic Interactions in Ferroelectric Crystals-I:Single Solitons and Domain Walls, Phys. Rev., B30, 5306–5325 (1984).
J. Pouget and G.A. Maugin, Influence of an Electric Field on the Motion of a Ferroelectric Domain Wall, Phys.Lett., 109A, 389–392 (1985).
G.A.Maugin and A. Miled, Solitary Waves in Elastic Ferromagnets, Phys.Rev.B, (submitted for publication in,1985).
G.A.Maugin and A. Miled, Solitary Waves in Micropolar Elastic Crystals, Int.J.Eng. Sci. (submitted for publication in,1985).
J.Pouget and G.A.Maugin, Solitary Waves in Oriented Elastic Crystals (being completed).
G.A.Maugin and J. Pouget, Solitons in Microstructured Elastic Media:Physical and Mechanical Aspects, in Continuum Models of Discrete Systems (5), Ed.A.J.M. Spencer, A.A.Balkema, Amsterdam (1985).
G.A. Maugin, Elastic-Electromagnetic Resonance Couplings in Electromagnetically Ordered Crystals, in Theoretical and Applied Mechanics, Eds. F.P.J. Rimrott and B. Tabarrok, pp. 345–355, North-Holland, Amsterdam (1980).
G. A. Maugin, Symmetry Breaking and Electromagnetic-Elastic Couplings, in The Mechanical Behavior of Electromagnetic Solid Continua, Ed. G.A. Maugin, pp. 35–46, North Holland, Amsterdam (1984).
J. Pouget, Influence de champs rémament ou initiaux sur les propriétés dynamiques de milieux élastiques polarisables, Doctoral Thesis in Mathematics, Université Pierre-et-Marie Curie, Paris, France (Mars 1984).
V.I. Karpman and E.M. Maslov, Perturbation Theory for solitons, Sov. Phys,. J.E.T.P., 46, 281–291 (1977).
J.P. Keener and D.W. McLaughlin, Solitons under Perturbations, Phys.Rev., A16, 777–790 (1977).
V.I. Karpman and E.M. Maslov, A Perturbation Theory for the Korteweg-deVries Equation, Phys.Lett., 60A, 307–308 (1977).
R.K. Bullough, P.J. Caudrey and H.M. Gibbs, in Solitons, Vol. 17 of Topics in Current Physics, pp. 107–141, Springer-Verlag, Berlin (1980).
A.C. Newell, Synchronized Solitons, J.Math.Phys., 18, 922–926 (1977).
A.C. Scott, in Backlund Transformations, the Inverse Scattering Method, Solitons and their Applications, Vol 515 of Lecture Notes in Mathematics, Ed. R.Miura, pp. 80–105, Springer-Verlag, Berlin (1976).
R. Hirota, Direct Methods in Soliton Theory, in Solitons, Vol. 17 of Topics in Current Physics, pp. 157–176, Springer-Verlag, Berlin (1980).
D.W. McLaughlin and A.C. Scott, Perturbation Analysis of Fluxon Dynamics, Phys.Rev., A18, 1652–1680 (1978).
R.K. Dodd, J.C. Eilbeck, J.D. Gibbon and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, London (1982).
J. Pouget and G.A. Maugin, Solitons and Electroacoustic Interactions in Ferroelectric Crystals-II:Interactions of Solitons and Radiations, Phys.Rev., B31, 4633–4649 (1985).
G.B. Whitham, Linear and Nonlinear Waves, J.Wiley-Interscience, New York (1974).
G.A.Maugin and J.Pouget, Transient Motion of a Solitary Wave in Elastic Ferroelectrics, Proc.Intern.Conf.Nonlinear Mechanics, Shanghai, China, Oct.28–31(1985).
J.Pouget, in these proceedings.
V.I. Karpman, Soliton Evolution in the Presence of Perturbation, Physics Scripta, 20, 462–478 (1979).
S. Motogi and G.A. Maugin, Effects of Magnetostriction on Vibrations of Bloch and Néel Walls, Physics statu solidi, a81, 519–532 (1984).
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Maugin, G. (1986). Solitons and domain structure in elastic crystals with a microstructure. In: Kröner, E., Kirchgässner, K. (eds) Trends in Applications of Pure Mathematics to Mechanics. Lecture Notes in Physics, vol 249. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0016392
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DOI: https://doi.org/10.1007/BFb0016392
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