Abstract
Discrete nonlinear lattice systems have attracted considerable interest in the last years.
It is well established that nonlinear lattice systems may exhibit selflocalized excitations
in form of solitons or breathers which are spatially localized and time-periodic solutions.
In the present work we discuss wave transmission and localization properties in a system
of two coupled onedimensional nonlinear chains. The equations we used to discribe the models
are discrete nonlinear Schrödinger equations. Their study is done with a dynamical
systems approach.
Nonlinearity and discreteness conspire into producing localized modes as well as global lattice
properties which do not exist in continous models.
Many investigations have been performed to explore the stationary and dynamical properties
of self-localized states, but most studies focused on systems extending in one spatial
direction only. This thesis have the aim to investigate the properties and dynamics
of two coupled discrete nonlinear chains, which are built up by an infinite set of nonlinear
oszillators distributed in space.
There are different possible discretizations of the continuum nonlinear Schrödinger
equation as a model of significant physical relevance. All these different discretizations
are nonintegrable in the case of modelling a double chain of coupled nonlinear oszillators.
Therefore numerical simulations have been an important tool to investigate great sets of
nonintegrable differential equations.
The physical contexts of the nonlinear Schrödinger equation are ranging from optical pulse
propagation in nonlinear fibres to condensed matter physics, fluid mechanics and biophysics.
We undertake a detailed discussion of the stationary properties of the generalized discrete
nonlinear Schrödinger equation (GDNLS) which interpolates between the discrete selftrapping
equation (DST) and the Ablowitz-Ladik equation (AL), which are discretizations of the
continuum nonlinear Schrödinger equation as well.
We apply the Melnikov method to get localized stationary excitations on the double chain, which are
stable in space and time. Therefore we study the homoklinc behaviour nearby hyperbolic fixpoints
of the coresponding nonlinear map, which we get from a stationary ansatz.
Homoklinic chaos of the map is the prior condition to find stable oszillator amplitude distributions
on the double chain. Furthermore we discuss in detail the energy transfer properties of a moving localized
excitation between the coupled chains.
We focus on the parameter dependence of the energy transfer and
investigate the coupling constitution which provides a maximal energy exchange between the
onedimensional lattices with the aim to model donator-acceptor systems consisting of two different
coupled nonlinear chains. |
