Solitons as a signature of the modulation instability in the discrete nonlinear Schrodinger equation

Abstract

The effect of the modulation instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schr\"odinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves is a signature of the modulation instability is used to analytically study instability effects on solitons during propagation. In particular, we concern with instability effects in the dark region, where other analytical methods as the standard modulation analysis of planewaves do not provide any information on solitons. The region of high-velocity solitons is studied anew showing that solitons are less prone to intabilities in this region. An analytical upper boundary for the self-defocusing instability is defined.