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Physical and Mathematical Models of Nonlinear Waves in Solids

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Nonlinear Waves in Solids

Part of the book series: CISM Courses and Lectures ((CISM,volume 341))

Abstract

In a first part (Sections 1 to 4), continuum and discrete (lattice) models of solids such as in elasticity and anelasticity are introduced with special attention paid to nonlinearity and dispersion. This is extended to solids with a microstructure of mechanical or electromagnetic origin. The second part (Sections 5 to 7) exemplifies models and properties of nonlinear-wave problems with an emphasis on solitary waves and soli-tons. Both exactly integrable and nearly integrable systems are considered. Systems governed by sine-Gordon, Boussinesq, Korteweg-de Vries, nonlinear Schrödinger and Zakharov equations or systems belong to the first class. Generalized Boussinesq, Zakharov and sine-Gordon-d’Alembert systems belong to the second class. The main properties of such systems are illustrated by computer-generated figures. Energy and pseudomomentum balances are presented as useful tools in such studies. Solitonic and dissipative structures are discriminated.

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References

  1. Engelbrecht, J.: Nonlinear Wave Processes of Deformation in Solids, Pitman, London 1983.

    Google Scholar 

  2. Eringen, A.C. and Maugin, G.A.: Electrodynamics of Continua, Vol.I and II, Springer-Verlag, New York 1990.

    Google Scholar 

  3. Eringen, A.C. and Suhubi, E.S.: Elastodynamics, Vol. I, Academic Press, New York 1975.

    Google Scholar 

  4. Maugin, G.A.: Nonlinear Electromechanical Effects and Applications - A Series of Lectures, World Scientific, Singapore 1985.

    Google Scholar 

  5. Maugin, G.A.: Continuum Mechanics of Electromagnetic Solids, North-Holland, Amsterdam 1988.

    Google Scholar 

  6. Maugin, G.A.:The Thermomechanics of Plasticity and Fracture, Cambridge University Press, U.K. 1992.

    Google Scholar 

  7. Maugin, G.A.: Material Inhomogeneities in Elasticity, Chapman and Hall, London 1993.

    Google Scholar 

  8. Maugin, G.A. and Trimarco, C.: Pseudomomentum and material forces in nonlinear elasticity: variational formulations and applications to brittle fracture, Acta Mechanica, 94 (1992), 1–28.

    Google Scholar 

  9. Maugin, G.A.: The principle of virtual power in continuum mechanics: Application to coupled fields, Acta Mechanica, 35 (1980), 1–70.

    Google Scholar 

  10. Maugin, G.A., Pouget, J., Drouot, R. and Collet B.: Nonlinear Electromechanical Couplings, J.Wiley, Chichester and New York 1992.

    Google Scholar 

  11. Bui, H.D.: Introduction aux Problèmes Inverses en Mécanique des Matériaux, Eyrolles, Paris 1993.

    Google Scholar 

  12. Maugin, G.A. and Muschik, W.: Thermodynamics with internal variables-I General concepts, II-Applications, J.Non-Equilibr.Thermodynam., 18 (1993)

    Google Scholar 

  13. Maugin, G.A.: Internal variables and dissipative structures, J.Non-Equilibr.Thermodynam., 15 (1990) 173–192.

    Google Scholar 

  14. Askar, A.: Lattice Dynamical Foundations of Continuum Theories, World Scientific, Singapore 1986.

    Google Scholar 

  15. Kosevich, A.M.: Theory of Crystal Lattices (in Russian), Vishtsha Shkola, Kiev 1988.

    Google Scholar 

  16. Kunin, I.A.: Elastic Media with Microstructure: One-Dimensional Models, Springer-Verlag, Berlin 1982.

    Google Scholar 

  17. Mindlin, R.D.: Elasticity, piezoelectricity and crystal lattice dynamics, J.Elasticity, 2 (1972), 217–282.

    Google Scholar 

  18. Askar A.: A model for coupled rotation-displacement modes of certain molecular crystals. Illustration for KNO, J.Phys.Chem.Solids, 34 (1973), 1901–1907.

    CAS  Google Scholar 

  19. Pouget, J., Askar, A. and Maugin, G.A.: Lattice model for elastic ferroelectric crystals:Microscopic Approach, Phys.Rev., 33 (1986), 6304–6319.

    CAS  Google Scholar 

  20. Pouget, J. Askar, A. and Maugin, G.A.: Latice model for elastic ferroelectric crystals: Continuum approximation, Phys.Rev., 33 (1986), 6320–6325.

    CAS  Google Scholar 

  21. Maugin, G.A. and Pouget, J.: Electroacoustic equations for one-domain ferroelectric bodies, J.Acoust.Soc.Amer., 68, (1980), 575–587.

    Google Scholar 

  22. Erofeev, V.I. and Potarov A.I.: Nonlinear wave processes in elastic media with microstructurg, in: Nonlinear World, pp. 1197–1215, Proc.Phys., Kiev 1990.

    Google Scholar 

  23. Pouget, J. and Maugin G.A.: Nonlinear dynamics of oriented elastic solids-I, II, J.Elasticity, 22 (1989), 135–155, 157–183.

    Google Scholar 

  24. Maugin, G.A. and Pouget, J.: Solitons in microstructured elastic media–Physical and mechanical models, in: Continuum Models of Discrete Systems (5), ed. A.J.M. Spencer, pp. 115–137, Bakema, Rotterdam 1987.

    Google Scholar 

  25. Maradudin, A.A., Montroll, B.W., Weiss, G.H.,and Ipatova I.P.: Theory of Lattice Dynamics in the Harmonic Approximation, Academic Press, New York 1971.

    Google Scholar 

  26. Weiner, J.H.: Statistical Mechanics of Elasticity, J.Wiley-Interscience, New York 1983.

    Google Scholar 

  27. Jeffrey, A. and Kawahara, T.: Asymptotic Methods in Nonlinear Wave Theory, Pitman, Boston 1982.

    Google Scholar 

  28. Cadet, S.: Dynamique non linéaire des réseaux atomiques: solitons transversaux et couplage avec les mouvements longitudinaux, Doct. Thesis in Physics, Univ. de Bourgogne, Dijon, France 1987.

    Google Scholar 

  29. Flytzanis, N., Pnevmatikos, St. and Remoissenet, M.: Kink, breather and asymmetric envelope or dark solitons in nonlinear chains: I.Monoatomic chain, J.Phys.C: Solid State Phys., 18 (1985), 4603–4629.

    CAS  Google Scholar 

  30. Pnveumatikos, St.: Solitons in nonlinear atomic chains, in: Singularities and Dynamical Systems, ed. S.N.Pnevmatikos, pp. 397–437, Elsevier, Amsterdam 1985.

    Google Scholar 

  31. Gorbacheva, O.B. and Ostrovsky, L.A.: Nonlinear vector waves in a mechanical model of molecular chain, Physica, 8 (1983), 223.

    Google Scholar 

  32. Collet,B. and Pouget, J.: Propagation d’ondes acoustiques dans les réseaux cristallins anharmoniques, Acustica, 63 (1987), 265–275.

    Google Scholar 

  33. Maugin, G.A.: Solitons in elastic crystals exhibiting phase transitions, in: Nonclassical continuum mechanics: Abstract Techniques and Applications, Eds. R.Knops and A.A.Lacey, pp. 272–283, Cambridge University Press, U.K. 1987.

    Google Scholar 

  34. Pouget, J.: Nonlinear dynamics of lattice models for elastic continua, in:Physical properties and thermodynamical behaviour of minerals, ed. K.Saljé, pp.359–402, Reidel, Dordrecht 1988.

    Google Scholar 

  35. Pouget, J.: Dynamics of patterns in ferroelastic-martensitic transformations I- Lattice model, II-Quasi-continuum, Phys.Rev., B43 (1991), 3575–3581, 3582–3592.

    Google Scholar 

  36. Maugin, G.A. and Cadet, S.: Existence of solitary waves in martensitic alloys, Int.J.Engng.Sci., 29 (1991), 243–258.

    CAS  Google Scholar 

  37. Christov, C.I. and Maugin, G.A.: A numerical venture into the menagerie of coherent structures of generalized Boussinesq systems, in: Coherent structures in Physics and Biology, Eds. M.Remoissenet and M.Peyrard, pp. 209–216, Springer-Vrelag, Berlin 1991.

    Google Scholar 

  38. Frenkel, J. and Kontorova, T.: On the theory of plastic deformation and twinning, Phys.Sowjet Union, 13 (1938), 1.

    CAS  Google Scholar 

  39. Barone, A., Esposito, F., Magee, G.J. and Scott, A.C.: Theory and applications of the sine-Gordon equation, Riv. Nuovo Cimento, 1 (1971), 227–267.

    Google Scholar 

  40. Weiner, J.H. and Sanders, W.T.: Peierls’ stress and creep of a linear chain, Phys.Rev., 134A (1964), 1007.

    Google Scholar 

  41. Eringen, A.C.: Theory of micropolar elasticity, in: Fracture, Vol.II, Chap.7, ed. H.Liebowitz, pp. 621–729, Academic Press, New York 1968.

    Google Scholar 

  42. Maugin, G.A. and Miled A.: Solitary waves in micropolar elastic crystals, Int.J.Engng.Sci., 24 (1986), 1477–1499.

    Google Scholar 

  43. Erbay, S., Erbay, H.A. and Dost, S.: Nonlinear wave modulation in micropolar elastic media-I-Longitudinal waves, II-Transverse waves, Int.J.Engng.Sci., 29 (1991), 845–858.

    Google Scholar 

  44. Erbay, S., Erbay, H.H. and Dost, S.: Nonlinear wave interactions in a micropolar elastic medium,Wave Motion, 16 (1992), 163–172.

    Google Scholar 

  45. Flytzanis, N., Pnevmatikos, St. and Remoissenet, M.: Soliton Resonances in atomic nonlinear chains, Physica, 26 (1987), 311–320.

    Google Scholar 

  46. Pnevmatikos, St. Flytzanis, N. and Remoissenet, M.: Soliton dynamics of nonlinear diatomic lattices, Phys.Rev., B33 (1986), 2308–2321.

    CAS  Google Scholar 

  47. Euvrard, D.: Résolution numérique des équations aux dérivées partielles (différences finies, éléments finis), Masson, Paris 1988.

    Google Scholar 

  48. Fermi, E., Pasta, J. and Ulam, S.: Studies of nonlinear problems-I, Los Alamos Report LA1940, New Mexico, 1955.

    Google Scholar 

  49. Richtmyer, R.D. and Morton, K.W.: Difference methods for initial value problems, 2nd Edition, Interscience, New York 1967.

    Google Scholar 

  50. Berezin, Yu.A.: Modelling non-linear wave processes (translation from the Russian), VNU Press, Utrecht 1987.

    Google Scholar 

  51. Rosenau, P.: Dynamics of nonlinear mass-spring chains near the continuum limit, Phys.Lett., 118A (1986), 222–227.

    Google Scholar 

  52. Ostrovskii, L.A. and Suttin, A.M.: Nonlinear waves in rods, P.M.M., 41 (1977), 543–549.

    Google Scholar 

  53. Soerensen, M.P., Christiansen, P.L. and Lomdahl, P.S.: Solitary waves in nonlinear elastic rods-I, J.Acoust.Soc.Amer., 76 (1984), 871–879.

    Google Scholar 

  54. Iskandar, L. and Jain, P.C.: Numerical solutions of the improved Boussinesq equation, Proc. Indian Acad.Sci. (Math.Sci.), 89 (1980), 171–181.

    Google Scholar 

  55. Bogolubsky, I.L.: Some examples of inelastic soliton interaction, Comput.Phys.Commun., 13 (1977), 149–155.

    Google Scholar 

  56. Clarkson, P.A., LeVeque, R.J. and Saxton, R.: Solitary wave interactions in elastic rods, Studies in Appl. Math., 75 (1986), 95–122.

    Google Scholar 

  57. Kruskal, M.D. and Zabusky, N.J.: J.Math.Phys., 5 (1964), 231.

    Google Scholar 

  58. Christov, C.I. and Maugin, G.A.: Numerics of some generalized models of lattice dynamics (Higher-order nonlinear and triple interactions), in: Nonlinear waves in solids, Eds. J.L.Wegner and F.Norwood, ASME, New York 1993.

    Google Scholar 

  59. Christov, C.I. and Maugin, G.A.: Long-time evolution of acoustic signals in nonlinear crystals, in: Proc.l3th Intern.Symp. on Nonlinear Acoustics, ed. H.Hobaek, World Scientific, Singapore 1993.

    Google Scholar 

  60. Samsonov, A.M.: On the existence of longitudinal strain solitons in an infinite nonlinearly elastic rod, Sov.Phys.Doklady, 299 (1988), 1083–1086.

    Google Scholar 

  61. Samsonov, A.M.: Nonlinear acoustic strain waves in elastic wave guides, in: Frontiers of Nonlinear Acoustics (Proc.l2th ISNA, Austin), Eds. M.F.Hamilton and D.T.Blackstock, pp. 583–588, Elsevier, London 1990.

    Google Scholar 

  62. Samsonov, A.M. and Sokurinskaya, E.V.: Energy exchange between nonlinear waves in elastic wave guides and external media, in: Nonlinear waves in active media, ed. J.Engelbrecht, pp. 99–104, Springer-Verlag, Berlin 1989.

    Google Scholar 

  63. Nelson, D.F.: Electric, Optic and Acoustic Interactions in Dielectrics, J.Wiley-Interscience, New York 1979.

    Google Scholar 

  64. Maugin, G.A.: Non-equilibrium thermodynamics of electromagnetic solids, in: Non-equilibrium thermodynamics with applications to solids (CISM Lecture Notes, 1992), ed. W.Muschik, Springer-Verlag, Wien 1993.

    Google Scholar 

  65. Ani, W. and Maugin, G.A.: Basic equations for shocks in nonlinear electroacoustic materials, J.Acoust.Soc.Amer., 85 (1989), 599–610.

    Google Scholar 

  66. Boulkeroua, A.: Lattice-dynamics theory of elastic crystals with a molecular group - Application to NaNO (in French), Doct.Thesis in Mechanics, U.P.M.C., Paris 1985.

    Google Scholar 

  67. Pouget,J. and Maugin, G.A.: Solitons and electroacoustic interactions in ferroelectric crystals -I- Single solitons and domain walls, Phys.Rev., B30 (1985), 5306–5325.

    Google Scholar 

  68. Sayadi, M.K.: Nonlinear dynamics of coherent structures in materials with microstructure (in French),Doct.Thesis in Mechanics, U.P.M.C., Paris 1991.

    Google Scholar 

  69. Pouget, J. and Sayadi, M.K.: Electromechanical coherent structures for ferroelectric elastic crystals, in: Mechanical modelling of new electromagnetic materials, ed. R.K.T. Hsieh, pp. 179–188, Elsevier, Amsterdam 1990.

    Google Scholar 

  70. Sayadi, M.K. and Pouget, J.: Soliton dynamics in a microstructured lattice model, J.Phys.A.Gen.Phys., 24 (1991), 2151–2172.

    Google Scholar 

  71. Soumahoro, K.: Mechanical behavior of antiferroelectric crystals, Doct.Thesis in Mechanics, U.P.M.C., Paris 1988.

    Google Scholar 

  72. Zorski, H. and Infeld, E.: New soliton equation for dipole chains, Phys.Rev.lett., 68 (1992), 1180–1183.

    Google Scholar 

  73. Maugin, G.A. and Miled, A.: Solitary waves in elastic ferromagnets, Phys.Rev., B33 (1986), 4830–4842.

    Google Scholar 

  74. Kosevich, A.M.: Dynamical and topologival solitons in ferromagnets and antiferromagnets, in: Solitons, Eds. S.E. Trullinger, V.E. Zakharov and V.L. Pokrovsky pp. 555–603, Elsevier, Amsterdam 1986.

    Google Scholar 

  75. Kosevich, A.M., Ivanov, B.A. and Kovalev, A.S.: Nonlinear Waves in Magnetic systems: Dynamical and topological solitons (in Russian), Naukova Dumka, Kiev 1983.

    Google Scholar 

  76. Daher, N.: Principle of virtual power: Application to nonlinear electroacoustics, State Doct.Thesis in Physics, University of Besançon, France 1987.

    Google Scholar 

  77. Daher, N. and Maugin, G.A.: Deformable semiconductors with interfaces: basic equations, Int.J.Engng.Sci., 25 (1987), 1093–1129.

    Google Scholar 

  78. Daher, N. and Maugin, G.A.: Nonlinear electroacoutsic equations in semiconductors with interfaces, Int.J.Engn.Sci., 26 (1988), 37–58.

    Google Scholar 

  79. Maugin, G.A.: Irreversible thermodynamics of deformable superconductors, C.R.Acad.Sci.paris, II-314 (1992), 889–994.

    Google Scholar 

  80. Bland, D.R.: Nonlinear dynamic elasticity, Blaisdell, Waltham, Mass. 1969

    Google Scholar 

  81. Varley, E.(editor): Propagation of shock waves in solids, Vol.AMD 17, A.S.M.E., New York 1976.

    Google Scholar 

  82. Wright, T.W.(editor): Nonlinear wave propagation in mechanics, Vol. AMD 77, A.S.M.E., New York 1986.

    Google Scholar 

  83. Graham, R.A.: Solids under high pressure shock compression, Springer-Verlag, Berlin 1993.

    Google Scholar 

  84. Brun, L.: Ondes de choc finies dans les solides élastiques, in: Mechanical waves in solids, Eds. J.mandel and L.Brun, pp. 63–155, Springer-Verlag, Wien 1975.

    Google Scholar 

  85. Perrin, G. and Delannoy-Coutris, M.: Analysis of plane elastic-plastic shock waves from the fourth-order anharmonic theory, Mechanics of Materials, 2 (1983), 139–153.

    Google Scholar 

  86. Piau, M.: Lois de comportement et ondes dans les milieux viscoplastiques en déformation finie, State Doct.Thesis in Mathematics, Université Pierre et Marie Curie, Paris 1975.

    Google Scholar 

  87. Raniecki, B.: Ordinary waves in inviscid plastic media, in: Mechanical waves in solids, Eds.J.Mandel and L.Brun, pp. 157–219, Springer-Verlag, Berlin 1975.

    Google Scholar 

  88. Nowacki, W.K.: Stress waves in non-elastic solids (translation from the Polish), Pergamon Press, Oxford 1978.

    Google Scholar 

  89. Engelbrecht, J.: An introduction to asymmetric solitary waves, Longman, London 1991.

    Google Scholar 

  90. Calogero,F. and Degasperis, A.: Spectral Transforms and Solitons-I, North-Holland, Amsterdam 1982.

    Google Scholar 

  91. Ablowitz, M.S. and Segur, H.: Solitons and the inverse scattering transform, SIAM, Philadelphia 1981.

    Google Scholar 

  92. Newell, A.C.: Solitons in mathematics and physics, SIAM, Philadelphia 1985.

    Google Scholar 

  93. Hasegawa, A.: Optical solitons in fibers, Springer-Verlag, Berlin 1989.

    Google Scholar 

  94. Zakharov, V.E. and Shabat, A.B.: Exact theory of two-dimensional self-focusing and one-dimensional self-modulation in nonlinear media, Soviet Phys. J.E.T.P., 34 (1972), 62–69.

    Google Scholar 

  95. Hui, C.Y. and Riedel, H.: The asymptotic stress and strain field near the tip of a growing crack under creep conditions, Int.J.Fracture, 17 (1981), 409.

    Google Scholar 

  96. Cadet, S.: Transverse envelope solitons in an atomic chain, Phys.Lett., 121 (1987), 77–82.

    CAS  Google Scholar 

  97. Cadet, S.: Propagation and interactions of nonlinear shear waves in a discrete lattice, Wave Motion, 11 (1989), 77–97.

    Google Scholar 

  98. Cadet, S.: Coupled transverse-longitudinal envelope modes in an atomic chain, Wave Motion, 11 (1989).

    Google Scholar 

  99. Pouget, J.: Nonlinear dynamics and instability of twin boundaries in martensitic-ferroelastic materials, in: Continuum models and discrete systems, ed. G.A.Maugin, Vol.I, pp. 296–312, Longman, London 1990.

    Google Scholar 

  100. Christov, C.I. and Maugin, G.A.: An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system, J.Comput.Phys. (submitted for publication 1993 ).

    Google Scholar 

  101. Christov, C.I. and Maugin, G.A. (in preparation, 1993 )

    Google Scholar 

  102. Ewen, J., Gunshor, R.L. and Weston, V.H.: Solitons in surface acoustic wave media, Proc. 1981 Ultrasonics Symposium, p.295–298, I.E.E.E., New York 1981.

    Google Scholar 

  103. Cantrell, J.H.: Nonlinear phenomena in solid state physics and technology, Proc. 1990 Ultrasonics Symposium, I.E.E.E., New York 1990.

    Google Scholar 

  104. Kielczynski, P.: Nonlinear ultrasonic methods in materials characterization (a Review), Internal report IMC-NRCC, Boucherville, PQ, Canada, March 1991

    Google Scholar 

  105. Kalyanasundaram, N.: Nonlinear surface acoustic waves on anisotropic solids, Int.J.Engng.Sci., 19 (1981), 279–286.

    Google Scholar 

  106. Lardner, R.W.: Nonlinear surface waves on an elastic solid, Int.J.Engng.Sci., 21 (1983), 1331–1342.

    Google Scholar 

  107. Planat, M.: Muttiple-scale analysis of the nonlinear acoustic wave propagation in anisotropic crystals, J.Appl.Phys., 57 (1985), 4911–4915.

    Google Scholar 

  108. Parker, D.F. and Talbot, F.M.: Analysis and computation for nonlinear elastic surface waves of permanent form, J.Elasticity, 15 (1985), 389–426.

    Google Scholar 

  109. Parker, D.F.: Wave form evolution for nonlinear surface elastic waves, Int.J.Engn.Sci., 26 (1988), 59–75.

    Google Scholar 

  110. Harvey, A.P. and Tupholme, G.E.: Propagation of anisotropic elastic and piezo-electric nonlinear surface waves, Wave Motion, 16 (1992), 125–136.

    Google Scholar 

  111. Bataille, K. and Lund, F.: Nonlinear waves in elastic media, Physica, 6D (1982), 95–104.

    Google Scholar 

  112. Maradudin, A.A.: Nonlinear surface acoustic waves and their associated surface acoustic solitons, in: Recent developments on Surface Acoustic Waves, Eds. D.F.Parker and G.A.Maugin, pp. 62–71, Springer-Verlag, Berlin 1988.

    Google Scholar 

  113. Maradudin, A.A. and Mayer, A.P.: Surface acoustic waves on nonlinear substrates, in: Nonlinear waves in solid states physics, Eds. A.D.Boardman, M.Bertolotti and T.Twardowski, pp. 113–161, Plenum, New York 1991.

    Google Scholar 

  114. Hadouaj, H. and Maugin, G.A.: Une onde solitaire se propageant sur un substrat élastique recouvert d’un film mince, C.R.Acad.Sci.Paris, II-309 (1989), 1877–1881.

    Google Scholar 

  115. MAUGIN, G.A. and Hadouaj, H.: Solitary surface transverse waves on an elastic substrate coated with a thin film, Phys.Rev., B44 (1991), 1266–1280.

    Google Scholar 

  116. Maugin, G.A., Hadouaj, H. and Malomed, B.A.: Nonlinear coupling between SH surface solitons and Rayleigh modes on elastic structures, Phys.Rev., B45 (1992), 9688–9694.

    Google Scholar 

  117. Hadouaj, H. and Maugin, G.A.: Surface solitons on elastic structures: numerics, Wave Motion, 16 (1992), 115–125.

    Google Scholar 

  118. Kalyanasundaram, N.: Finite-amplitude Love-waves on an isotropic layered half-space, Int.J.Engng.Sci., 19 (1981), 287–293.

    Google Scholar 

  119. Teymur, M.: Nonlinear modulation of Love waves ina compressible hyperelastic layered half space, Int.J.Engng.sci., 26 (1988), 907–927.

    Google Scholar 

  120. Gorentsveig, V.I., Kivshar, Yu.S., Kosevich, A.M. and Syrkin, E.S.: Nonlinear surface elastic modes in crystals, Phys.Lett., 144 (1990), 479–486.

    Google Scholar 

  121. Kivshar, Yu.S.: Solitons in a nonlinear elastic medium, Phys.Rev., B43 (1991)

    Google Scholar 

  122. Collet, B.: Solitons acoustiques guidés dans une plaque cristalline à symétrie cubique, J.Physique Coll.C1, Suppl.III, 2 (1992), C1–733 - C1–736.

    Google Scholar 

  123. Collet, C.: Lattice dynamics approach of Shear horizontal solitons in cubic crystal plates, in: Continuum Models of Discrete Systems (7), ed. K.-H. Anthony, Trans.Techn.Publ. Switz. 1993.

    Google Scholar 

  124. Daher, N. and Maugin, G.A.: The method of virtual power in continuum mechanics: Application to media presenting singular surfaces and interfaces, Acta Mechanica, 60 (1986), 217–240.

    Google Scholar 

  125. Murdoch, A.I.: The propagation of surface waves in bodies with material boundaries, J.Mech.Phys.Solids, 24 (1976), 137–146.

    Google Scholar 

  126. Mozhaev, V.G.: A new type of acoustic waves in solids due to nonlinearity, Phys.Lett., A139 (1989), 333–337.

    Google Scholar 

  127. Whitham, G.B.: Linear and nonlinear waves, J.WileyInterscience, New York 1974.

    Google Scholar 

  128. Benney, D.J. and Newell, A.C.: The propagation of nonlinear wave envelopes, J.Math. and Phys (now Stud.Appl.Math), 46 (1967), 133–139.

    Google Scholar 

  129. Zakharov, V.E.: Collapse of Langmuir waves, Soviet Phys.J.E.T.P., 35 (1972), 908–914.

    Google Scholar 

  130. Hadouaj, H., Malomed, B.A. and Maugin, G.A.: Dynamics of a soliton in the generalized Zakharov’s system, Phys.Rev., A44 (1991), 3925–3931.

    CAS  Google Scholar 

  131. Hadouaj, H., Malomed, B.A. and Maugin, G.A.: Solitonsoliton collisions in the generalized Zakharov’s system, Phys.Rev., A44 (1991), 3932–3940.

    CAS  Google Scholar 

  132. Maugin, G.A., Hadouaj, H. and Malomed, B.A.: Some properties of Soliton solutions of the generalized Zakharov system, (Proc.Intern.Conf.Waves and Stability, Acireale, 1990), Le Matematiche, XLVI (1991), 253–264.

    Google Scholar 

  133. Kivshar, Yu.S. and Malomed, B.A.: Dynamics of solitons in nearly integrable systems, Rev.Modern Physics, 61 (1989), 763–915.

    Google Scholar 

  134. Wesolowski, Z.: Dynamics of a bar of asymmetric cross-section, J.Engng.Math., 17 (1983), 315–321.

    Google Scholar 

  135. Engelbrecht, J., Peipman, P. and Valdeck, U.: Nonlinear effects of acoustics in solids, in: Frontiers of Nonlinear Acoustics (Proc.l2th ISNA,1990), Eds. M.K. Hamilton and D.T. Blackstock, pp. 535–540, Elsevier, London 1990.

    Google Scholar 

  136. Engelbrecht, J. and Peipman, P.: Nonlinear waves in a layer with energy influx, Wave Motion, 16 (1992), 173–181.

    Google Scholar 

  137. Rassokha, A.: Soliton localization of plastic deformation, Contributed paper to XVIIth ICTAM, Haifa, Israel, Aug.22–28, Abstracts p.122, IUTAM, 1992.

    Google Scholar 

  138. Ani, W. and Maugin, G.A.: One-dimensional nonlinear motions in electroelastic solids, Z.angew.Math.Phys., 39 (1988), 277–298.

    Google Scholar 

  139. Collet, B.: Nonlinear wave propagation in elastic dielectrics with internal variables, J.Techn.Phys., 26 (1985), 285–289.

    Google Scholar 

  140. Collet, B.: Transient nonlinear waves in deformable dielectric materials, in: Electromagnetomechanical interactions in deformable solids and structures, Eds. Y.Yamamoto and K.Miya, pp. 329–334, North-Holland, Amsterdam 1987.

    Google Scholar 

  141. Maugin, G.A., Collet,B. and Pouget, J.: Electromechanical waves in ceramics–Numerical simulations, Appl.Math.and Mech.(Shanghai), 6 (1986), 1129–1139.

    Google Scholar 

  142. Maugin, G.A., Collet, B. and Pouget, J.: Nonlinear wave propagation in coupled electromechanical systems, in: Nonlinear wave propagation in mechanics, ed. T.W.Wright, Vol.AMS 77, pp. 57–83, A.S.M.E., New York 1986.

    Google Scholar 

  143. Kalyanasundaram, N.: Nonlinear propagation characteristics of Bleustein-Gulyaev waves; J.Sound Vibr., 96 (1984), 411–420.

    Google Scholar 

  144. Tupholme, G.E. and Harvey, A.P.: Nonlinear surface acoustic waves on a piezoelectric solid, Int.J.Engng. Sci., 26 (1988), 1161–1166.

    Google Scholar 

  145. Harvey, A.P and Tupholme, G.E.: Nonlinear mode coupling of two co-directional surface acoustic waves on a piezoelectric solid, Int.J.Engng.Sci., 29 (1991), 987–998.

    Google Scholar 

  146. Parker, D. F. and David, E. A.: Nonlinear piezoelectric surface waves, Int.J.Engng.Sci., 27 (1989), 565–581.

    Google Scholar 

  147. David, E.A. and Parker D.F.: Nondistorting wave forms of electroelastic surface waves, Wave Motion, 12 (1990), 987–998.

    Google Scholar 

  148. Maugin, G.A.: Wave motion in magnetizable deformable solids, Int.J.Engng.Sci., 19 (1981), 321–388.

    Google Scholar 

  149. Pouget, J. and Maugin, G. A.: Solitons and electroacoustic interactions in ferroelectric crystals–II. Interactions of solitons and radiations, Phys.Rev., B31 (1985), 4633–4649.

    CAS  Google Scholar 

  150. Maugin, G.A.: Solitons in elastic crystals with a microstructure–Mathematical aspects, in: Trends in Applications of Pure Mathematics to Mechanics, Eds. E.Kröner and K.Kirchgassner, pp. 195–211, Springer-Verlag, Berlin 1986.

    Google Scholar 

  151. Pouget, J. and Maugin, G.A.: Influence of an external electric field on the motion of a ferroelectric domain wall, Phys.Lett., 109A (1985), 389–392.

    Google Scholar 

  152. Christiansen, P.L. and Oslen, O.H.: Propagation of fluxons on Josephson lines with impurities, Wave Motion, 4 (1982), 163–172.

    Google Scholar 

  153. Kivshar, Yu.S. and Malomed, B.A.: Dynamics of domain walls in elastic ferrmagnets and ferroelectrics, Phys. Rev., B42 (1990), 8561–8570.

    Google Scholar 

  154. Planat, M.: L’instabilité des ondes se propageant dans un milieu piézoélectrique, Coll.Phys.Suppl.C3, 51 (1990), C3–207 - C3–217.

    Google Scholar 

  155. Planat, M.: Observation of acoustic envelope solitary waves generated by metallic interdigital transducers on a quartz crystal, in: Continuum models and Discrete systems, ed. G.A.Maugin, Vol.2, pp. 339–349, Longman, London 1991.

    Google Scholar 

  156. Maugin, G.A. and Engelbrecht, J.: A Thermodynamical viewpoint to nerve-pulse dynamics, J.Non-Equilibr. Thermodyn., (in the press, 1993 ).

    Google Scholar 

  157. Barenblatt, G.I.: Intermediate asymptotics,(in Russian) 2nd Edition, Meteoizdrat, Moscow 1982.

    Google Scholar 

  158. Vavilev, V.A., Romanovskii, Yu.M, Chernayskii, D.S., and Yakhno, V.G.: Autowave processes in kinetic systems, V.E.B. Deutscher verlag der Wissen., Berlin 1987.

    Google Scholar 

  159. Sachdev, P.L.: Nonlinear diffusive waves, Cambridge University Press, U.K, 1987.

    Google Scholar 

  160. Maugin, G.A.: Applications of an energy-momentum tensor in nonlinear elastodynamics: Pseudomomentum and Eshelby stress in solitonic elastic systems, J.Mech.Phys.Sol., 29 (1992), 1543–1558.

    Google Scholar 

  161. Maugin, G.A.: Analytical and numerical problems for nonlinear wave propagation in “nearly” integrable systems, in: Analytical and Numerical aspects of wave propagation, ed. R.E. Kleinman, S. I. A.M., Philadelphia 1993.

    Google Scholar 

  162. Fokas, A.S.: Generalized symmetries and constants of motion of evolution equations, Lett.Math.Phys., 3 (1979), 467–473.

    Google Scholar 

  163. Maugin,G.A.: On the balance of pseudo-momentum in the mechanics and electrodynamics of continua, C.R.Acad.Sci.Paris, 311 (1990), 763–768.

    Google Scholar 

  164. Malomed, B.A.: Inelastic interactions of solitons in nearly integrable systems-I, II, Physica, D15 (1985), 374–384, 385–401.

    Google Scholar 

  165. Sander, J. and Hutter, K.: On the development of the theory of the solitary wave. A historical essay, Acta Mechanica, 86 (1991), 111–152.

    Google Scholar 

  166. Newell A.C.: The history of the soliton, Trans.ASME.J. Appl.Mech., 50 (1983), 1127–1138.

    Google Scholar 

  167. Scott-Russel, J., Boussinesq, J. and Appell, P.: L’onde solitaire (a collection of short texts and three engravings) by Marc Pessin, Artist-engraver, (150 signed copies only), Le Verbe et l’Empreinte, Saint-Laurent du Pont, Isère, France 1987.

    Google Scholar 

  168. Jammer, M.: The Philosophy of quantum mechanics, J. Wiley - Interscience, New York 1974.

    Google Scholar 

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Authors and Affiliations

  1. Pierre et Marie Curie University, Paris, France

    G. A. Maugin

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Editors and Affiliations

  1. University of Newcastle Upon Tyne, UK

    A. Jeffrey

  2. Estonian Academy of Sciences, Estonia

    J. Engelbrecht

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Maugin, G.A. (1994). Physical and Mathematical Models of Nonlinear Waves in Solids. In: Jeffrey, A., Engelbrecht, J. (eds) Nonlinear Waves in Solids. CISM Courses and Lectures, vol 341. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2444-4_3

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  • DOI: https://doi.org/10.1007/978-3-7091-2444-4_3

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