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Oscillatory wave fronts in chains of coupled nonlinear oscillators

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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Abstract

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.

Item Type:Article
Uncontrolled Keywords:Semiconductor superlattices; Discrete; Propagation; Dynamics; Failure; Systems
Subjects:Sciences > Physics > Mathematical physics
Sciences > Mathematics > Differential equations
ID Code:15039
References:

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[23] 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 → �}∞.

[24] 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.

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Last Modified:06 Feb 2014 10:14

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