Quasilinear dynamics in nonlinear evolution equations

Vadim Zharnitsky, Mathematics, UIUC

Nonlinear systems usually produce nonlinear dynamics except for some very special cases, such as small amplitude solutions or perturbation of a solitary wave. We describe another mechanism (which we call quasilinear) for linear behavior in strongly nonlinear systems. We demonstrate the phenomenon for nonlinear Schoedinger equation with periodic boundary conditions, where we have nearly complete understanding of the phenomenon. We also present numerical simulations for other equations where analytical methods have not, yet, been developed. We will discuss the origin and application of this phenomenon in the field of optical communication. This is joint work with M. Burak Erdogan.