Discrete Breathers And Travelling Waves Of Discrete Nonlinear Schr(?)dinger Equation

In the first part of this paper, we consider the existence and stability of discrete breathers of the nonlinear Schrodmger(DNLS) equation with ac-driving and damping in one-dirnensional lattices:Instead of using the commonly used homoclinic orbit approach, which can not give the rigorous analysis , we study in detail the existence and stability of the periodic solutions of the single oscillator. Then we give a continuation theorem of the zeros of a map defined in R×l^∞, and apply this theorem to the DNLS equation and prove that when the damping force 7, driving force h and frequency ω satisfy ω² > 3γ²,h₁² < h² < h₂², a discrete dark breather with the frequency ω exist. At last we show that the stability can be analyzed rigorously due to the detailed discussion of the single oscillator and the spectral theory of perturbation of operator.In the second part we consider the parametrically-driven DNLS equation without damping: By the transformation , where ω= Ω/2, and some proper scale transformations, the above system is converted towhere φ(ï¿¡) = η_n(t), ï¿¡ = —(n — vt). As shown in Ref.[27, 28], applying the center manifold theory in infinite dimensions in [29], the above equation can be reduced to a system of ordinary differential equations whose dimension equals the dimension of the invariant subspace belonging to the central part of the spectrum Σ₀ of the linearization at 0. Then for any given (α, r) (?) â–³₀ = {(a,r)| E₀L_α.r contains only one pair of simple eigenvalues ±iq₁}, the system reduces to a two dimensional smooth vector field. But it's not a reversible system, so we have the small amplitude travelling waves η_n(t) = φ[—(n — vt)] by the Poincare — Andronov — Hopf bifurcation theorem.