Department of Theoretical Physics,

National Institute of Physics and Nuclear Engineering,

Magurele, P.O.Box MG-6, 76900 Bucharest, ROMANIA

E-mail: avisin@theory.nipne.ro, avisin@ifin.nipne.ro

**Modulational instability and multiple scales analysis
of Davydov's model**

**Abstract:**

The modulational instability (MI) of some discrete nonlinear evolution
equations (DNEE), representing approximations of Davydov's model of alpha-helix
in protein, is studied. In a multiple scales analysis the dominant amplitude
usually satisfies the nonlinear Schroedinger equation (NLS), or Zakharov-Benney
equations (ZB) if a long wave-short wave resonance takes place. The MI
is studied both from a deterministic and statistical point of view. If
the second derivative of the linear dispersion relation is positive (focusing
case of NLS) the system is unstable at small modulations of the amplitude.
In the statistical approach, when the field variable is considered as a
random field, a new phenomenon, similar to the Landau damping in plasma
physics, can appear. This is carefully analized for a Lorentzian distribution
of the unperturbed Fourier transform of the two-point correlation function.

**References:**

- A.S. Davydov, "Soliton in Molecular Systems" (Reidel, Dordrecht, 1985).
- R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, "Solitons and Nonlinear Wave Equations" (Acad. Press, 1982).
- I.E. Alber, Proc. Roy. Soc. London A 363, 525 (1978).
- D. Grecu, Anca Visinescu, Physics AUC v. 12, 129-149 (2002).