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Instability of solitary waves in nonlinear composite media. (English. Russian original) Zbl 1036.74029
J. Appl. Math. Mech. 65, No. 6, 977-984 (2001); translation from Prikl. Mat. Mekh. 65, No. 6, 1008-1016 (2001).
The authors studies plane wave motions in inhomogeneous nonlinear elastic medium (composite). It is assumed that the displacements \(w_\alpha\), the strains \(u_\alpha = \partial w_\alpha/\partial x\) and the velocities of particles \(v_\alpha\) \((\alpha = 1,2,3)\) depend on a single Cartesian coordinate \(x_3\) and on time \(t\). Basic equations describing plane waves are presented, their Hamiltonian form is found, and symmetries in anisotropic and isotropic cases are studied. The authors also discuss the instability of solitary waves.

MSC:
74J35 Solitary waves in solid mechanics
74E30 Composite and mixture properties
74H55 Stability of dynamical problems in solid mechanics
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References:
[1] Bakhvalov, N. S.; Eglit, M. E.: Variational properties of the averaged equations of periodic media. Tr. mat. Inst. akad nauk 192, 5-19 (1990) · Zbl 0780.73045
[2] Bakhvalov, N. S.; Eglit, M. E.: Effective equations with dispersion for the propagation of waves in periodic media. Dokl. ross. Akad. nauk 370, No. 1, 7-10 (2000) · Zbl 1047.35015
[3] Il’ichev, A. T.: Stability of solitons in non-linear composite media. Zh. eksper. Teor. fiz. 118, No. 3, 720-729 (2000)
[4] Il’ichev, A.: Stability of solitary waves in nonlinear composite media. Physica D 150, 264-277 (2001) · Zbl 1013.74044
[5] Kulikovskii, A. G.; Sveshnikova, Ye.I.: Non-linear waves in elastic media. (1998)
[6] Gvozdovskaya, N. I.; Kulikovskii, A. G.: Quasitransverse shock waves in elastic media with an internal structure. Zh. prikl. Mekh. tekh. Fiz. 40, No. 2, 174-180 (1999) · Zbl 0945.74040
[7] Sibgatullin, N. R.: Non-linear transverse oscillations accompanying resonance in an elastic layer of an ideally conducting fluid. Prikl. mat. Mekh. 36, No. 1, 79-87 (1972)
[8] Grillakis, M.; Shatah, J.; Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. Funct. anal. 74, 160-197 (1987) · Zbl 0656.35122
[9] Bakholdin, I. B.: The structures of evolutional jumps in reversible systems. Prikl. mat. Mekh. 63, No. 1, 52-62 (1999) · Zbl 1019.76008
[10] Bakholdin, I.; Il’ichev, A.: Radiation and modulational instability described by the fifth-order Korteweg-de Vries equation. Contemporary mathematics 200, 1-15 (1996) · Zbl 0861.76011
[11] Bakholdin, I. B.; Zharkov, A. A.; Il’ichev, A. T.: Decay of solitons in an isothermal, collision-free, quasi-neutral plasma with an isothermal pressure. Zh. eksper. Teor. fiz. 118, No. 1, 125-141 (2000)
[12] Beliaev, A.; Il’ichev, A.: Conditional stability of solitary waves propagating in elastic rods. Physica D 90, 107-118 (1996) · Zbl 0899.73282
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