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Solitons in elastic solids (1938–2010). (English) Zbl 1272.74364

Summary: Solitons in solids are much less studied than in fluids or in optics although the seminal Fermi-Pasta-Ulam numerical experiment and its interpretation by Kruskal et al. indeed belong to this framework. An inquisitive observer and then an active participant for almost forty years, the author presents here the various developments that took place over this period in the solid mechanics and dynamics of lattices and/or structural members, as also the original results that followed thereby. Most of the solutions obtained deviate from standard ones as the physical systems deduced from first principles generally are not exactly integrable. The emphasis is placed on the peculiarities of the solutions in terms of analytical expressions, their interpretation, and their eventual representation as quasi-particles in steady (inertial) or accelerated motion.

MSC:

74J35 Solitary waves in solid mechanics
74-02 Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids
74-03 History of mechanics of deformable solids
01A60 History of mathematics in the 20th century
01A61 History of mathematics in the 21st century
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[1] Ablowitz, M. J.; Segur, H.: Solitons and the inverse scattering transform, (1981) · Zbl 0472.35002
[2] Aero, E. L.; Kuvshinskii, E. V.: Fundamental equations of the theory of elastic media with rotationally interacting particles (Engl. Transl.), Sov. phys. Solid state 2, 1272-1281 (1961)
[3] Aizu, K.: Possible species of ferromagnetic, ferroelectric and ferroelastic crystals, Phys. rev. 2, 754-772 (1970)
[4] Bäcklund, A. V.: Zur theorie der flächentransformationen, Math. ann. 19, 387-422 (1882) · JFM 13.0296.01
[5] Barone, A.; Esposito, F.; Magee, C. G.; Scott, A. C.: Theory and applications of the sine-Gordon equation, Riv. nuovo cimento 1, 227-267 (1971)
[6] Bataille, K.; Lund, F.: Nonlinear waves in elastic media, Physica D 6, 95-104 (1982)
[7] Benjamin, T. B.; Bona, J. L.; Mahony, J. J.: Model equation for long waves in nonlinear dispersive systems, Phil. trans. R. soc. Lond. A 272, 47-78 (1972) · Zbl 0229.35013
[8] Benney, D. J.; Newell, A. C.: The propagation of nonlinear wave envelopes, J. math. Phys. (now, stud. Appl. math.) 46, 133-139 (1967) · Zbl 0153.30301
[9] Bogdan, M. M.; Kosevich, A. M.; Maugin, G. A.: Formation of soliton complexes in dispersive systems, Condens. matter phys. 2, No. 18, 255-265 (1999)
[10] Bogdan, M. M.; Kosevich, A. M.; Maugin, G. A.: Soliton complex dynamics in strongly dispersive systems, Wave motion 34, 1-26 (2001) · Zbl 1074.35572
[11] Boussinesq, J. V.: Théorie nouvelle des ondes lumineuses, J. math. Pures et appliquées 2, No. 13, 313-339 (1870) · JFM 01.0362.01
[12] Cadet, S.: Propagation and interactions of nonlinear shear waves in a discrete lattice, Wave motion 11, 77-97 (1989)
[13] Cho, Y.; Miyagawa, N.: Surface acoustic wave solitons propagating on the metallic grating waveguide, Appl. phys. Lett. 63, 1188-1190 (1993)
[14] Christov, C. I.; Maugin, G. A.: An implicit difference scheme for the long-time evolution of localized solutions of a generalized Boussinesq system, J. comput. Phys. 116, 39-51 (1995) · Zbl 0822.65118
[15] Christov, C. I.; Maugin, G. A.: Numerics of some generalized models of lattice dynamics (higher-order nonlinear and triple interactions), Nonlinear waves in solids 137, 374-378 (1995)
[16] Christov, C. I.; Maugin, G. A.; Velarde, M. G.: Well-posed Boussinesq paradigm with purely spatial higher-order derivatives, Phys. rev. E 54, 3621-3638 (1996)
[17] Christov, C. I.; Maugin, G. A.; Porubov, A. V.: On Boussinesq’s paradigm in nonlinear wave propagation, special issue on J.V. Boussinesq, CR mécanique (Acad. Sci. Paris) 335, No. 9/10, 521-535 (2007) · Zbl 1134.74028
[18] Collet, B.: Lattice approach for shear horizontal solitons in cubic crystal elastic plates, Mater. sci. Forum 123-125, 417-426 (1993)
[19] Courant, R.; Friedrichs, K. O.: Supersonic flow and shock waves, (1948) · Zbl 0041.11302
[20] Drazin, P. G.; Johnson, R. S.: Solitons: an introduction, (1989) · Zbl 0661.35001
[21] Eckl, C.; Schöllman, J.; Mayer, A. P.; Kovalev, A. S.; Maugin, G. A.: On the stability of surface acoustic pulse trains in coated elastic media, Wave motion 34, 35-49 (2001) · Zbl 1074.74565
[22] Ewen, J.; Gunshor, R. L.; Weston, V. H.: Solitons in surface acoustic waves, , 295-298 (1981)
[23] Ewen, J.; Gunshor, R. L.; Weston, V. H.: An analysis of solitons in surface acoustic wave devices, J. appl. Phys. 53, 5682 (1982)
[24] Falk, F.: Ginzburg – Landau theory of static domain walls in shape-memory alloys, Zeit. phys. C: condens. Matter 51, 177-185 (1983)
[25] Fokas, A. S.: Generalized symmetries and constants of motion of evolution equations, Lett. math. Phys. 3, 467-473 (1979) · Zbl 0426.35051
[26] Frenkel, J.; Kontorova, T.: On the theory of plastic deformation and twinning, Physik. sowjetunion 123, 1-15 (1938) · JFM 64.1422.02
[27] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M.: Method for solving the Korteweg – de Vries equation, Phys. rev. Lett. 19, 1095-1097 (1967) · Zbl 1103.35360
[28] Gorentsveig, V. I.; Kivshar, Yu.S.; Kosevich, A. M.; Syrkin, E. S.: Nonlinear surface modes in crystals, Phys. lett. 144, 479-486 (1990) · Zbl 0762.73020
[29] Hadouaj, H.; Maugin, G. A.: Surface solitons on elastic structures: numerics, Wave motion 16, 115-125 (1992) · Zbl 0764.73023
[30] Hadouaj, H. A.; Malomed, B. A.; Maugin, G. A.: Dynamics of a soliton in the generalized Zakharov’s system, Phys. rev. A 44, 3925-3931 (1991)
[31] Hadouaj, H. A.; Malomed, B. A.; Maugin, G. A.: Soliton-soliton collisions in the generalized Zakharov’s system, Phys. rev. A 44, 3932-3940 (1991)
[32] Hadouaj, H.; Maugin, G. A.; Malomed, B. A.: Nonlinear coupling between SH surface solitons and Rayleigh modes on elastic structures, Phys. rev. B 45, 9688-9694 (1992)
[33] Infeld, E.; Rowlands, G.: Nonlinear waves, solitons and chaos, (1990) · Zbl 0726.76018
[34] Kivshar, Y. S.; Malomed, B. A.: Dynamics of solitons in nearly integrable systems, Rev. mod. Phys. 61, 763-915 (1989)
[35] Korteweg, D. J.; De Vries, G.: On the change of form of long waves advancing in a rectangular canal, and on new type of long stationary wave, Philos. mag. 39, No. 5, 422-443 (1895) · JFM 26.0881.02
[36] Kosevich, A. M.: The discrete lattice: phonons, solitons, dislocations, (1999)
[37] Kosevich, A. M.; Ivanov, B. A.; Kovalev, A. S.: Nonlinear magnetization waves – dynamical and topological solitons, (1988)
[38] Kovalev, A. S.; Syrkin, E. S.; Maugin, G. A.: Many-dimensional and surface solitons in nonlinear elastic media, J. low temp. Phys. 28, No. 6, 635-647 (2002)
[39] Kovalev, A. S.; Mayer, A. P.; Eckl, C.; Maugin, G. A.: Solitary Rayleigh waves in the presence of surface nonlinearities, Phys. rev. E 66, No. 3, 036615-36621 (2002)
[40] Kovalev, A. S.; Sokolova, E. S.; Mayer, A. P.; Maugin, G. A.: Nonlinear Rayleigh waves in half space covered with atomic monolayer, Low temp. Phys. 29, 530-536 (2003)
[41] Kovalev, A. S.; Zolotarev, D. A.; Maugin, G. A.: Nonlinear surface spin waves in ferromagnets, Phys. met. Metall. 95, No. Suppl. 1, S35-S40 (2003)
[42] Kovalev, A. S.; Gerasimchuk, I. V.; Maugin, G. A.: Nonlinear dynamics of incommensurate surface layers, Phys. rev. Lett. 92, 244101-1-244101-4 (2004)
[43] Kruskal, M. D.; Zabusky: Exact invariants for a class of nonlinear wave equations, J. math. Phys. 7, 1265-1267 (1966) · Zbl 0171.07201
[44] Maradudin, A. A.: D.f.parkerg.a.mauginnonlinear surface acoustic waves and their associated solitons, Nonlinear surface acoustic waves and their associated solitons, 62-71 (1988)
[45] Maradudin, A. A.; Mayer, A. P.: Surface acoustic waves on nonlinear susbtrates, Nonlinear waves in solid state physics, 13-161 (1991)
[46] Maugin, G. A.: Continuum mechanics of electromagnetic solids, (1988) · Zbl 0652.73002
[47] Maugin, G. A.: Pseudomomentum in solitonic elastic systems (In the honour of P. Chadwick, FRS, Dublin, November 1991), J. mech. Phys. solids 40, 1543-1558 (1992) · Zbl 0791.73017
[48] Maugin, G. A.: Material inhomogeneities in elasticity, (1993) · Zbl 0797.73001
[49] Maugin, G. A.: On some generalizations of Boussinesq and KdV systems, Proc. acad. Sci. Estonia (Special issue: KdV equation) A 44, 40-55 (1995) · Zbl 0844.35092
[50] Maugin, G. A.: Nonlinear waves in elastic crystals (Book in the series ”Oxford monographs in mathematics”), (1999) · Zbl 0943.74002
[51] Maugin, G. A.: Theory of nonlinear surface waves and solitons, Surface waves in geomechanics (Six lectures at Udine, September 2004), 325-371 (2005) · Zbl 1094.74032
[52] Maugin, G. A.: Nonlinear surface waves and solitons, Eur. phys. J. special topics 147, 209-230 (2007)
[53] Maugin, G. A.; Cadet, S.: Existence of solitary waves in martensitic alloys, Int. J. Eng. sci. 29, 243-255 (1991) · Zbl 0762.73018
[54] Maugin, G. A.; Christov, C. I.: Nonlinear waves and conservation laws (Nonlinear duality between elastic waves and quasi-particles), Selected topics in nonlinear wave mechanics, 117-160 (2002) · Zbl 1026.74041
[55] Maugin, G. A.; Hadouaj, H.: Une onde solitaire se propageant sur un substrat élastique recouvert d’un film mince, CR acad. Sci. Paris II 309, 1877-1881 (1989) · Zbl 0684.73014
[56] Maugin, G. A.; Hadouaj, J.: Solitary surface transverse waves on an elastic substrate coated with a thin film, Phys. rev. B 44, 1266-1280 (1991)
[57] Maugin, G. A.; Miled, A.: Solitary waves in elastic ferromagnets, Phys. rev. B 33, 4830-4842 (1986) · Zbl 0613.73004
[58] Maugin, G. A.; Miled, A.: Solitary waves in micropolar elastic crystals, Int. J. Eng. sci. 24, 1477-1499 (1986) · Zbl 0613.73004
[59] Mayer, A. P.: Surface acoustic waves in nonlinear elastic media, Phys. rep. 256, 237-366 (1995)
[60] Mozhaev, V. G.: A new type of acoustic waves in solids due to nonlinearity, Phys. lett. A 139, 333-337 (1989)
[61] Murdoch, A. I.: The propagation of surface waves in bodies with material boundaries, J. mech. Phys. solids 24, 137-146 (1976) · Zbl 0342.73017
[62] Nayanov, V. I.: Surface acoustic cnoïdal waves and solitons in a linbo3-(SiO film), JETP lett. 44, 314-317 (1986)
[63] Newell, A. C.: Solitons in mathematics and physics, (1985) · Zbl 0565.35003
[64] Ostrovsky, L. A.; Johnson, P. A.: Dynamic nonlinear elasticity in geomaterials, Riv. nuovo cimento 24, No. 7, 1-46 (2001)
[65] , Recent developments in surface acoustic waves, vol.7 of Springer series on wave phenomena (Proc. EURMECH coll. No. 226, 2 – 5 September 1987, Nottingham, UK) 7 (1988)
[66] Pnevmatikos, St.; Flytzanis, N.; Remoissenet, M.: Soliton dynamics in nonlinear diatomic lattices, Phys. rev. B 33, 2308-2321 (1986)
[67] Porubov, A. V.: Amplification of nonlinear strain waves in solids, (2003) · Zbl 1058.74005
[68] Porubov, A. V.: Localization of nonlinear waves of deformation, (2009) · Zbl 1189.74073
[69] Porubov, A. V.; Maugin, G. A.: Longitudinal strain solitary waves in presence of cubic nonlinearity, Int. J. Non-linear mech. 40, 1041-1048 (2005) · Zbl 1349.74213
[70] Porubov, A. V.; Maugin, G. A.: Propagation of localized longitudinal strain waves in a plate in presence of cubic nonlinearity, Phys. rev. E 74, 046617-46621 (2006)
[71] Porubov, A. V.; Maugin, G. A.: Improved description of longitudinal strain solitary waves, J. sound vib. 310, No. 3, 694-701 (2008)
[72] Porubov, A. V.; Maugin, G. A.: Cubic nonlinearity and longitudinal surface solitary waves, Int. J. Non-linear mech. 44, 552-559 (2009)
[73] Porubov, A. V.; Maugin, G. A.; Mareev, V. V.: Localization of two-dimensional nonlinear train waves in a plate, Int. J. Non-linear mech. 39, No. 8, 1359-1370 (2004) · Zbl 1348.74177
[74] Porubov, A. V.; Aero, E. L.; Maugin, G. A.: Two approaches to study essentially nonlinear and dispersive properties of the internal structure of materials, Phys. rev. E 79, No. 4 (2009)
[75] Potapov, A. I.; Pavlov, I. S.; Gorshkov, K. A.; Maugin, G. A.: Nonlinear interactions of solitary waves in a 2D lattice, Wave motion 34, 83-96 (2001) · Zbl 1074.74588
[76] Pouget, J.: Nonlinear dynamics of lattice models for elastic continua, NATO summer school on physical properties and thermodynamical behavior of minerals, Oxford, 359-402 (1988)
[77] Pouget, J.: Dynamics of patterns in ferroelastic – martensitic transformations. I. lattice model. II. quasi-continuum, Phys. rev. B 43 (1991)
[78] Pouget, J.; Maugin, G. A.: Solitons and electroacoustic interactions in ferroelectric crystals. I. single solitons and domain walls, Phys. rev. B 30, 5306-5325 (1984)
[79] Pouget, J.; Maugin, G. A.: Solitons and electroacoustic interactions in ferroelectric crystals. II. interactions between solitons and radiations, Phys. rev. B 31, 4633-4651 (1985)
[80] Pouget, J.; Maugin, G. A.: Influence of an external electric field on the motion of a ferroelectric domain wall, Phys. lett. A 109, 389-392 (1985)
[81] Pouget, J.; Maugin, G. A.: Nonlinear dynamics of oriented elastic solids. II. propagation of solitons, J. elasticity 22, 157-183 (1989) · Zbl 0697.73018
[82] Salupere, A.; Maugin, G. A.; Engelbrecht, J.: KdV soliton detection from a harmonic input, Phys. lett. A 192, 5-8 (1994) · Zbl 0959.35506
[83] Salupere, A.; Engelbrecht, J.; Kalda, J.; Maugin, G. A.: On the KdV soliton formation and discrete spectral analysis, Wave motion 23, 49-66 (1996) · Zbl 0968.76524
[84] Salupere, A.; Engelbrecht, J.; Maugin, G. A.: Solitons in systems with a quartic potential and higher-order dispersion (Proc. EUROMECH 348, Tallinn, May 1996), Proc. est. Acad. sci. Math. phys. 46, 118-127 (1997) · Zbl 0908.35120
[85] Salupere, A.; Engelbrecht, J.; Maugin, G. A.: Solitonic structures in KdV-based higher order systems, Wave motion 34, 51-61 (2001) · Zbl 1074.35576
[86] Samsonov, A.M., 2001. Strain Solitons in Solids, and How to Construct Them. Chapman & Hall/CRC, Boca Raton, Florida (This book contains a large bibliography on nonlinear waves in structural members, especially rods and plates; see the pioneering works by Nariboli, G.A., Sedov, A., 1970. Burgers – KdV equation for viscoelastic rods and plates. J. Math. Anal. Appl., 32, 661 – 677; Ostrovsky, L.A., Sutin, A.M., 1977. Nonlinear elastic waves in rods. Priklad. Matem. i Mekhan., 41/3, 531 – 537 (in Russian); Soerensen, M.P., Christiansen, P.L., Lomdahl, P.S., 1984. Solitary waves in nonlinear elastic rods I. J. Acous. Soc. Amer., 76, 871 – 879).
[87] Samsonov, A. M.; Dreiden, G. V.; Porubov, A. V.; Semenova, I. V.: Generation and observation of longitudinal strain soliton in a plate, Tech. phys. Lett. 22, 61-68 (1996)
[88] Sayadi, M. K.; Pouget, J.: Propagation d’excitations acoustiques non linéaires dans LES matériaux dotés de microstructure, J. phys. Coll. 51, No. C3, 219-230 (1990)
[89] Sayadi, M.; Pouget, J.: Soliton dynamics in a microstucturd lattice model, J. phys. (UK) A: gen. Phys. 24, 2151-2172 (1991)
[90] Sayadi, M.; Pouget, J.: Chaos transition of a motion in microstructured lattice, Physica D 55, 259-268 (1992) · Zbl 0743.34069
[91] Seeger, A., 1949. Diploma Physik, T.U. Stuttgart. (Ph.D., 1951).
[92] Seeger, A.: Theorie der gitterfehlstellen, Handbuch der physik bd. 7, 383-665 (1955)
[93] Seeger, A.: Solitons in crystals, (1979) · Zbl 0555.73007
[94] Seeger, A.; Wesolowski, Z.: Standing wave solutions of the enneper equation (sine-Gordon equation), Int. J. Eng. sci. 19, 1535-1549 (1981) · Zbl 0537.35066
[95] Stoker, J. J.: Water waves, (1957) · Zbl 0078.40805
[96] Whitham, G. B.: Linear and nonlinear waves, (1974) · Zbl 0373.76001
[97] Wesolowski, Z.: Dynamics of a bar of asymmetric cross section, J. eng. Math. 17, 315-322 (1983) · Zbl 0523.73045
[98] Zabusky, N. J.; Kruskal, M. D.: Interactions of ”solitons” in a collisionless plasma and recurrence of initial states, Phys. rev. Lett. 15, 240-243 (1965) · Zbl 1201.35174
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