Carpio Rodríguez, Ana María and Bonilla , L.L. (2003) Oscillatory wave fronts in chains of coupled nonlinear oscillators. Physical Review E, 67 (5). ISSN 1539-3755
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Official URL: http://arxiv.org/pdf/cond-mat/0303576.pdf
Wave front pinning and propagation in damped chains of coupled oscillators are studied. There are two important thresholds for an applied constant stress F: for \F\<F(cd) (dynamic Peierls stress), wave fronts fail to propagate, for F(cd)<\F\<F(cs) stable static and moving wave fronts coexist, and for \F\>F(cs) (static Peierls stress) there are only stable moving wave fronts. For piecewise linear models, extending an exact method of Atkinson and Cabrera's to chains with damped dynamics corroborates this description. For smooth nonlinearities, an approximate analytical description is found by means of the active point theory. Generically for small or zero damping, stable wave front profiles are nonmonotone and become wavy (oscillatory) in one of their tails.
|Uncontrolled Keywords:||Semiconductor superlattices; Discrete; Propagation; Dynamics; Failure; Systems|
|Subjects:||Sciences > Physics > Mathematical physics|
Sciences > Mathematics > Differential equations
 J. Frenkel and T. Kontorova, J. Phys. USSR, 13, 1 (1938). O.M. Braun and Yu.S. Kivshar, Phys. Rep. 306, 1 (1998).
 F.R.N. Nabarro, Theory of Crystal Dislocations (Oxford University Press, Oxford, UK, 1967).
 L.I. Slepyan, Sov. Phys. Dokl. 26, 538 (1981).
 P. M. Chaikin and T. C. Lubensky, Principles of condensed matter physics (Cambridge University Press, Cambridge, 1995). Chapter 10.
 A. Carpio, L.L. Bonilla and G. Dell’Acqua, Phys. Rev. E, 64, 036204 (2001).
 L.L. Bonilla, J. Phys. Condensed Matter 14, R341 (2002).
 A.R.A. Anderson and B.D. Sleeman, Int. J. Bif. Chaos, 5, 63 (1995). A. Carpio and L.L. Bonilla, SIAM J. Appl. Math. 63, 619 (2002).
 J.P. Keener and J. Sneyd, Mathematical Physiology (Springer, New York, 1998). Chapter 9.
 D.R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge U.P., Cambridge, UK, 2002).
 C. Rebbi and J. Soliani (eds.), Solitons and Particles, (World Sci., Singapore, 1984).
 A. Carpio and L.L. Bonilla, Phys. Rev. Lett. 86, 6034 (2001) and SIAM J. Appl. Math. 63, 1056 (2003).
 R. Hobart, J. Appl. Phys. 36, 1948 (1965).
 V.L. Indenbom, Soviet Phys. - Crystallogr. 3, 193 (1959)
[Kristallografiya 3, 197 (1958)].
 J. W. Cahn, Acta Metallurgica 8, 554 (1960).
 J.R. King and S.J. Chapman, Eur. J. Appl. Math. 12, 433 (2001).
 G. F´ath, Physica D, 116, 176 (1998).
 W. Atkinson and N. Cabrera, Phys. Rev. 138, A763 (1965).
 J. H. Weiner, Phys. Rev. 136, A863 (1964).
 V.H. Schmidt, Phys. Rev. B 20, 4397 (1979).
 P.C. Bressloff and G. Rowlands, Physica D 106, 255 (1997).
 S. Flach, Y. Zolotaryuk and K. Kladko, Phys. Rev. E 59, 6105 (1999).
 M. Peyrard and M.D. Kruskal, Physica D 14, 88 (1984).
 In the conservative limit m → ∞, the oscillations at one of both tails of the wave front do not decay as n → }∞. This means that a slightly more general definition of wave front solution is needed in the conservative case: un(_ ) = w(n − c_ ) joins the neighborhoods of U1(F/A) and U3(F/A) (or viceversa) as n increases from −∞ to ∞. An infinitesimal amount of damping causes the oscillatory tails of the wave front to decay to either U1(F/A) or U3(F/A) as n → }∞.
 We would like to point out that the numerical solution of Eq. (2) should be calculated by using a scheme of sufficiently high order and small tolerance for the cases _ = 0 or _ > 0 small (typically a standard Rung-Kutta- Fehlberg scheme of orders 4/5 with tolerance 10−5 would do). Schemes with lower order or larger tolerances have been found to produce spurious traveling waves. For the purpose of checking the appropriatedness of a given numerical scheme, the exact solutions we have discussed are invaluable.
|Deposited On:||27 Apr 2012 11:17|
|Last Modified:||27 Apr 2012 11:17|
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