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