Instability of local deformations of an elastic rod

Physica D: Nonlinear Phenomena

Volumn 182, Issue 1-2, 2003, Pages 103-124

  Lafortune, S ^a   Lega, J ^a  

^a The University of Arizona   (United States)

Author keywords

Elastic rod; Evans function; Klein Gordon equations

Indexed keywords

COMPUTER SIMULATION; CONVERGENCE OF NUMERICAL METHODS; EIGENVALUES AND EIGENFUNCTIONS; ELASTICITY; FUNCTIONS; MATHEMATICAL OPERATORS; NONLINEAR EQUATIONS; PROBLEM SOLVING;

ELASTIC RODS; EVANS FUNCTION; KLEIN-GORDON EQUATIONS;

DEFORMATION;

EID: 1642338748     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(03)00125-8     Document Type: Article

References (26)

1. Goriely A., Tabor M., Nonlinear dynamics of filaments. II. Nonlinear analysis. Physica D. 105:1997;45-61.
2. Lega J., Goriely A., Pulses, fronts and oscillations of an elastic rod. Physica D. 132:1999;373-391.
3. Dangelmayr G., Knobloch E., The Takens-Bogdanov bifurcation with O(2)-symmetry. Phil. Trans. R. Soc. Lond. A. 322:1987;243-279.
4. Evans J.W., Nerve axon equations. IV. The stable and unstable impulse. Ind. Univ. Math. J. 24:1975;1169-1190.
5. Jones C.K.R.T., Stability of the travelling wave solution to the FitzHugh-Nagumo equation. Trans. AMS. 286:1984;431-469.
6. Yanagida E., Stability of fast traveling pulse solutions of the FitzHugh-Nagumo equations. J. Math. Biol. 22:1985;81-104.
7. Alexander J., Gardner R., Jones C., A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410:1990;167-212.
8. Pego R., Weinstein M., Eigenvalues, and instabilities of solitary waves. Phil. Trans. R. Soc. Lond. A. 340:1992;47-94.
9. Swinton J., The stability of homoclinic pulses: a generalisation of Evans's method. Phys. Lett. A. 163:1992;57-62.
10. Brigdes T.J., Derks G., Hodge duality and the Evans function. Phys. Lett. A. 251:1999;363-372.
11. Gardner R.A., Zumbrun K., The Gap Lemma and geometric criteria for instability of viscous shock profiles. Commun. Pure Appl. Math. LI:1998;0797-0855.
12. Kapitula T., Sandstede B., Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations. Physica D. 124:1998;58-103.
13. B. Sandstede, Stability of travelling waves, in: B. Fiedler (Ed.), Handbook of Dynamical Systems. II. Towards Applications, Elsevier, Amsterdam, 2002, pp. 983-1055.
14. Alexander J.C., Jones C.K.R.T., Existence and stability of asymptotically oscillatory double pulses. J. Reine Angew. Math. 446:1994;49-79.
15. Alexander J.C., Jones C.K.R.T., Existence and stability of asymptotically oscillatory triple pulses. Z. Angew. Math. Phys. 44:1993;189-200.
16. Kapitula T., The Evans function and generalized Melnikov integrals. SIAM J. Math. Anal. 30:1998;273-297.
17. Kapitula T., Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation. Physica D. 116:1998;95-120.
18. Kapitula T., Sandstede B., Edge bifurcations for near integrable systems via Evans function techniques. SIAM J. Math. Anal. 33:2002;1117-1143.
19. Kapitula T., Rubin J., Existence and stability of standing hole solutions to complex Ginzburg-Landau equations. Nonlinearity. 13:2000;77-112.
20. Bridges T.J., Derks G., Unstable eigenvalues and the linearization about solitary waves and fronts with symmetry. Proc. R. Soc. Lond. A. 455:1999;2427-2469.
21. Bridges T.J., Derks G., The symplectic Evans matrix, and the instability of solitary waves and fronts. Arch. Rat. Mech. Anal. 156:2001;1-87.
22. Grillakis M., Shatah J., Strauss W., Stability theory of solitary waves in the presence of symmetry: II. J. Funct. Anal. 94:1990;308-348.
23. P. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993.
24. E. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955, p. 104.
25. Li Y.A., Promislow K., The mechanism of the polarizational mode instability in birefringent fiber optics. SIAM J. Math. Anal. 31:2000;1351-1373.
26. Goriely A., Nizette M., Tabor M., On the dynamics of elastic strips. J. Nonlinear Sci. 11:2001;3-45.