Existence Of Discrete Breathers Of Discrete Nonlinear Schr(?)dinger Equation

In this paper, we consider the existence of bright and dark discrete breathers ofthe parametrically driven and damped discrete nonlinear SchrSdinger equation iψ_n(t)+2|ψ_n(t)|^2σψ_n(t)+α[ψ_n+1(t)+ψ_n-1(t)-2ψ_n(t)]=hψ_n^*(t)e^iΩt-iγψ_n(t)and the generalized discrete nonlinear SchrSdinger equation iψ_n(t)=-γ|ψ_n(t)|^2σψ_n(t)-[α+μ|ψ_n(t)|^2σ][ψ_n+1(t)+ψ_n-1(t)]in one-dimensional lattices.First of all, we study a nonlinear smooth invertible map M: R²â†’R² defined by(x, z)→(z, -x+2z+f(z)). If the stable and unstable manifolds of M exist, thesymmetry of them is very useful. Using the symmetry of the manifolds, we proved theexistence of homoclinic and heteroclinic orbits of the map M.Next, we discuss the discrete nonlinear Schrodinger equations. We acquire analgebraic equation ofφ_n by substituting the ansatzψ_n(t)=φ_ne^iwt. Letφ_n=x_n+iy_n,where x_n, Y_n∈R. We obtain an equation of x_n by choosing a non-zero real constantk such that y_n=kx_n. Choosing an appropriate function f(z), we have a map Massociated with the equation of x_n.Lastly, we show the existence of bright and dark discrete breathers associated withthe homoclinic and heteroclinic orbits of the map M. The rigorous conclusions of thispaper are not restricted to the weakly coupled system, which is different from thoseworks using the continuation theorem.