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Stability of a class of solitary waves in compressible elastic rods. (English) Zbl 1115.74339

Summary: We prove that the solitary waves of a model for nonlinear dispersive waves in cylindrical compressible hyperelastic rods are orbitally stable. This establishes that the shape of the wave is stable.

MSC:

74J35 Solitary waves in solid mechanics
35B35 Stability in context of PDEs
35Q51 Soliton equations
74J30 Nonlinear waves in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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[1] J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973.; J.D. Achenbach, Wave Propagation in Elastic Solids, North-Holland, Amsterdam, 1973. · Zbl 0268.73005
[2] Benjamin, B.; Bona, J.; Mahony, J., Phil. Trans. R. Soc. (London), 272, 47 (1972)
[3] Camassa, R.; Holm, D., Phys. Rev. Lett., 71, 1661 (1993)
[4] Constantin, A.; Escher, J., Ann. Sci. Norm. Sup. Pisa, 26, 303 (1998)
[5] Constantin, A.; Escher, J., Acta Math., 181, 229 (1998)
[6] Constantin, A.; McKean, H. P., Commun. Pure Appl. Math., 52, 949 (1999)
[7] A. Constantin, L. Molinet, Orbital stability of the solitary waves for a shallow water equation, preprint 1998.; A. Constantin, L. Molinet, Orbital stability of the solitary waves for a shallow water equation, preprint 1998. · Zbl 0984.35139
[8] A. Constantin, W. Strauss, Stability of peakons, Commun. Pure Appl. Math., in press.; A. Constantin, W. Strauss, Stability of peakons, Commun. Pure Appl. Math., in press. · Zbl 1049.35149
[9] Dai, H.-H., Acta Mech., 127, 293 (1998)
[10] H.-H. Dai, Y. Huo, Solitary shock waves and other travelling waves in general compressible hyperelastic rod, Proc. R. Soc. (London), in press.; H.-H. Dai, Y. Huo, Solitary shock waves and other travelling waves in general compressible hyperelastic rod, Proc. R. Soc. (London), in press. · Zbl 1004.74046
[11] R. Dodd, J. Eilbeck, J. Gibbon, H. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1984.; R. Dodd, J. Eilbeck, J. Gibbon, H. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1984.
[12] N. Dunford, J.T. Schwartz, Linear Operators. Part II: Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988.; N. Dunford, J.T. Schwartz, Linear Operators. Part II: Spectral Theory. Selfadjoint Operators in Hilbert Space, Wiley, New York, 1988. · Zbl 0635.47002
[13] Fokas, A. S.; Fuchssteiner, B., Phys. D, 4, 47 (1981)
[14] Grillakis, M.; Shatah, J.; Strauss, W., J. Funct. Anal., 74, 160 (1987)
[15] T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Springer Lecture Notes in Mathematics, vol. 448, 1975, pp. 25-70.; T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in: Spectral Theory and Differential Equations, Springer Lecture Notes in Mathematics, vol. 448, 1975, pp. 25-70.
[16] A.M. Samsonov, Nonlinear strain waves in elastic waveguides, in: Nonlinear Waves in Solids, CISM Courses and Lectures, Springer, Vienna, 1994, pp. 349-382.; A.M. Samsonov, Nonlinear strain waves in elastic waveguides, in: Nonlinear Waves in Solids, CISM Courses and Lectures, Springer, Vienna, 1994, pp. 349-382. · Zbl 0806.73018
[17] Souganidis, P.; Strauss, W., Proc. R. Soc. (Edinburgh), 114A, 195 (1990)
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