Studies On Discrete Breathers In Nonlinear Lattices

Discrete breathers, which are spatial localized and time periodic, are very common nonlinear excited modes in discrete nonlinear systems. They play an absolutely important role in properties of the systems, especially thermal properties. The problem of discrete breathers in nonlinear lattice systems is investigated in the present paper.The studies on discrete breathers involve theories and experiments. The theories consist of existence and stability of discrete breathers, mobility, non-exponential relaxation, and targeted energy transfer, etc., while the experiments refer to Josephson junction networks, coupled nonlinear optical waveguides, ultracold atoms in optical lattices, micromechanical cantilever arrays, antiferromagnetic layered structures, and localized atomic vibrations in molecules, and so on. As a simple periodic discrete nonlinear system, the studies on discrete breathers in a lattice would have a distinct theoretical value.Inheriting the FPU problem, a more general and complex model, mixed KG/FPU atomic chain, is investigated. By now, as studies strictly aiming at this model with little consideration, it is novel in some sense.Extended rotating wave approximation is used to cut off the sequences of Fourier expansion of the exact solution to prepare for the following steps. Thus, the original equations of motion are transformed to algebraic equations about the Fourier coefficients. And the effects of each system parameter and the frequency on the discrete breathers and their components in the system are investigated by numerous numerical calculations. The research approach and perspective as a whole are original innovations in some degree.In a one-dimensional mixed KG/FPU monatomic chain, the symmetry of discrete breathers with frequencies in the space under or above the linear spectrum is intensively affected by the cubic inter-site nonlinear potential. The two cubic potentials induce the static and the second-order harmonic parts of the discrete breathers with certain symmetry. Moreover, due to the fine analyses of localization length and the Fourier components, the discrete breathers close to the edge of linear spectrum have some similarities and relevances with linear plane wave solutions of the system. It is consistent with the results obtained by central manifold theorem or variational method. Meanwhile, it provides direct numerical evidences for RWA, multiple scale method, and method of separation of variables.However, there are many similar results between one-dimensional diatomic and mono atomic chains. But the situations in diatomic chains are more complicated than those in the mono atomic chains. According to different types of initial conditions, there are four types of discrete breathers corresponding to one frequency. And only under the initial conditions with symmetry or anti-symmetry vibrations of light atoms, the cubic inter-site potential would induce anti-symmetric static parts.In general, the results obtained in the mixed KG/FPU atomic lattices have a common sense. By choosing different combinations of coefficients of the potentials, a mixed KG/FPU atomic lattice is corresponding to different special lattices, such as FPU or KG lattices. And the results are also agree with those obtained by other researchers in these special models. However, the method used in the chapter 4 and 5 has wide applicability and can educe more abundant results.According to the effects of nonlinear potentials on symmetries and their strength distributions, one is able to predict the existing discrete breathers in a given system, which is real helpful to theoretical analysis and numerical calculation. Meanwhile, it provides theoretical foundations for artificially designing new materials and devices with given properties and functions.