Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators

A. Carpio

doi:10.1103/PhysRevE.69.046601

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Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators

A. Carpio^*

Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain

^*Email address: ana $_{-}$ carpio@mat.ucm.es

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Phys. Rev. E 69, 046601 – Published 12 April, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.046601

Abstract

We present a stability theory for kink propagation in chains of coupled oscillators and a different algorithm for the numerical study of kink dynamics. The numerical solutions are computed using an equivalent integral equation instead of a system of differential equations. This avoids uncertainty about the impact of artificial boundary conditions and discretization in time. Stability results also follow from the integral version. Stable kinks have a monotone leading edge and move with a velocity larger than a critical value which depends on the damping strength.

References (23)

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A. Carpio and L.L. Bonilla, Phys. Rev. E 67, 056621 (2003).
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G. Fáth, Physica D 116, 176 (1998); A. Carpio and L.L. Bonilla, Phys. Rev. Lett. 86, 6034 (2001).
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[c1] J. Frenkel and T. Kontorova, Phys. Z. Sowjetunion 13, 1 (1938) [J. Phys. (USSR) 1, 137 (1939)]; O.M. Braun and Yu.S. Kivshar, Phys. Rep. 306, 1 (1998).

[c2] F.R.N. Nabarro, Theory of Crystal Dislocations (Oxford University Press, Oxford, UK, 1967).

[c3] L.I. Slepyan, Sov. Phys. Dokl. 26, 538 (1981).

[c4] P.M. Chaikin and T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University Press, Cambridge, 1995), Chap. 10.

[c5] L.L. Bonilla, J. Phys.: Condens. Matter 14, R341 (2002).

[c6] A. Tonnelier, Phys. Rev. E 67, 036105 (2003); A. Anderson and B. Sleeman, Int. J. Bifurcation Chaos Appl. Sci. Eng. 5, 63 (1995).

[c7] J.P. Keener and J. Sneyd, Mathematical Physiology (Springer, New York, 1998), Chap. 9.

[c8] R. Hobart, J. Appl. Phys. 36, 1948 (1965).

[c9] E. Gerde and M. Marder, Nature (London) 413, 285 (2001); D.A. Kessler, 413, 260 (2001).

[c10] A. Carpio and L.L. Bonilla, Phys. Rev. E 67, 056621 (2003).

[c11] W. Atkinson and N. Cabrera, Phys. Rev. 138, A763 (1965).

[c12] O. Kresse and L. Truskinovsky, J. Mech. Phys. Solids 51, 1305 (2003).

[c13] G. Fáth, Physica D 116, 176 (1998); A. Carpio and L.L. Bonilla, Phys. Rev. Lett. 86, 6034 (2001).

[c14] M. Peyrard and M.D. Kruskal, Physica D 14, 88 (1984).

[c15] A. Carpio, L.L. Bonilla, A. Wacker, and E. Schöll, Phys. Rev. E 61, 4866 (2000).

[c16] A. Carpio (unpublished).

[c17] V.H. Schmidt, Phys. Rev. B 20, 4397 (1979).

[c18] P.C. Bressloff and G. Rowlands, Physica D 106, 255 (1997).

[c19] S. Flach, Y. Zolotaryuk, and K. Kladko, Phys. Rev. E 59, 6105 (1999).

[c20] P. Rosenau, Phys. Lett. A 118, 222 (1986).

[c21] M.J. Boussinesq, J. Math. Pures Appl. 17, 55 (1872).

[c22] J. Bafaluy and J.M. Rubi, Physica A 153, 129 (1988); 153, 147 (1988).

[c23] E. Schrödinger, Ann. Phys. (Leipzig) 44, 1053 (1914).

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Nonlinear stability of oscillatory wave fronts in chains of coupled oscillators

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