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Zhou, Yong

Stability of solitary waves for a rod equation. (English) Zbl 1046.35094

Chaos Solitons Fractals 21, No. 4, 977-981 (2004).
Summary: We consider a new rod equation derived recently by H.-H. Dai [Acta Mech. 127, No. 1–4, 193–207 (1998; Zbl 0910.73036)] for a compressible hyperelastic material. We prove that the smooth solitary waves to this rod equation are orbital stable.

Cited in 22 Documents

MSC:

35Q51 Soliton equations
35B35 Stability in context of PDEs
74H55 Stability of dynamical problems in solid mechanics
74J35 Solitary waves in solid mechanics

Citations:

Zbl 0910.73036
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\textit{Y. Zhou}, Chaos Solitons Fractals 21, No. 4, 977--981 (2004; Zbl 1046.35094)
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References:

[1] Benjamin, T. B.; Bona, J. L.; Mahony, J. J., Model equations for long waves in nonlinear dispersive systems, Philos. Trans. R. Soc. London Ser. A, 272, 1220, 47-78 (1972) · Zbl 0229.35013
[2] Camassa, R.; Holm, D., An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71, 1661-1664 (1993) · Zbl 0972.35521
[3] Constantin, A.; Escher, J., Well-posedness, global existence and blow-up phenomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math., 51, 475-504 (1998) · Zbl 0934.35153
[4] Constantin, A.; Escher, J., Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181, 229-243 (1998) · Zbl 0923.76025
[5] Constantin, A.; Strauss, W., Stability of peakons, Commun. Pure Appl. Math., 53, 603-610 (2000) · Zbl 1049.35149
[6] Dai, H.-H., Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech., 127, 1-4, 193-207 (1998) · Zbl 0910.73036
[7] Dai, H.-H.; Huo, Y., Solitary shock waves and other travelling waves in a general compressible hyperelastic rod, R. Soc. London Proc. Ser. A Math. Phys. Eng. Sci., 456, 1994, 331-363 (2000) · Zbl 1004.74046
[8] Dunford, N.; Schwartz, J. T., Linear operators, Part III. Spectral Operators (1988), John Wiley and Sons, Inc: John Wiley and Sons, Inc New York
[9] Grillakis, M.; Shatah, J.; Strauss, W., Stability theory of solitary waves in the presence of symmetry, Int. J. Funct. Anal., 74, 1, 160-197 (1987) · Zbl 0656.35122
[10] Li, Y.; Olver, P., Well-posedness and blow-up solutions for an integrable nonlinear dispersive model wave equation, J. Different. Equat., 162, 27-63 (2000) · Zbl 0958.35119
[11] Lopes, O., Stability of peakons for the generalized Camassa-Holm equation, Electron. J. Different. Equat., 5, 12 (2003) · Zbl 1088.35060
[12] Lopes, O., Nonlocal variational problems arising in long wave propagation, ESAIM Control Optim. Calc. Var., 5, 501-528 (2000) · Zbl 0969.35046
[13] McKean, H. P., Breakdown of a shallow water equation, Asian J. Math., 2, 4, 867-874 (1998) · Zbl 0959.35140
[14] Shkoller, S., Geometry and curvature of diffeomorphism groups with \(H^1\) metric and mean hydrodynamics, J. Funct. Anal., 160, 1, 337-365 (1998) · Zbl 0933.58010
[15] Souganidis, P. E.; Strauss, W. A., Instability of a class of dispersive solitary waves, Proc. R. Soc. Edinburgh Sect. A, 114, 3-4, 195-212 (1990) · Zbl 0713.35108
[16] Xin, Z.; Zhang, P., On the weak solution to a shallow water equation, Commun. Pure Appl. Math., 53, 1411-1433 (2000) · Zbl 1048.35092
[17] Xin, Z.; Zhang, P., On the uniqueness and large time behavior of the weak solutions to a shallow water equation, Commun. Part. Different. Equat., 27, 9-10, 1815-1844 (2002) · Zbl 1034.35115
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