Modulational instability in some nonlinear one-dimensional lattices and soliton generation

The modulational instability (MI), also known as the Benjamin-Feir instability is a generic phenomenon in continuum and discrete nonlinear physical systems. The phenomenon is generally believed to be responsible for the formation of robust localized coherent structures. A system of equations have been obtained for the first time by Zakharov in plasma physics (coupling between Langmuir oscillations and ion sound) and by Benney in a hydrodynamic context (coupling between gravity and capillary modes of surface waves) and are known as the Zakharov-Benney equations. In the 1-D case it was proved to be an integrable system. The main results of this paper are the following: One shows that a Stokes wave solution of the ST-DNEE is unstable at small modulation of the amplitude if ω2>0, ω(k) being the dispersion relation and ω2 its second derivative. A multiple scales analysis of this model leads to the NLS equation for the dominant amplitude and ω2>0 corresponds to the focusing case when NLS equation admits solitonic solutions. The instability is related to the generation of sidebands around the principal wave and their resonant interaction. The problem of MI for the NLS equation is discussed also from a statistical point of view, when the amplitude is considered as a random variable. A kinetic equation for the Fourier transform of the two-point correlation function (with respect to the relative coordinate) is deduced. The linear stability analysis gives a stability equation similar to the dispersion relation obtained in the treatment of linearized Vlasov equation in plasma physics. The equation is solved both for a Gaussian and a Lorentzian form of the initial condition F0(k). Again the instability is present if ω2>0. The coupled system of the DNEE was discussed when a long-wave-short wave resonance takes place. The multiple scales method leads in this case to the Zakharov-Benney equations, which in 1-D are completely integrable. The MI for ZB equations is discussed from the deterministic point of view. (authors)