i Preface

The work presented in this Thesis is the result of close collaboration with several scientists, and the main part of the work has been published before in the articles 1{17], and the conference proceedings 18{26] or is going to be published 27]. The most important 13 articles 1{13] form the core of the Thesis. The Thesis consists of a review, the purpose of which is to relate the 13 articles and summarize the results. Finally, the 13 articles will be given in their complete form. Abstract This Thesis deals with nonlinear and stochastic dynamics in systems which can be described by nonlinear Schrr odinger models. Basically three diierent models are investigated. The rst is the continuum nonlinear Schrr odinger model in one and two dimensions generalized by a tunable degree of nonlinearity. Various perturbations such as uctuating forces, localized impurities, and dissipation are considered. The second kind of model is the discrete nonlinear Schrr odinger model, which also is investigated in one and two dimensions. External perturbations similar to the continuum case are also considered in the discrete model. Finally, nonlinear Schrr odinger continuum and discrete models with nonlocal dispersion are investigated. In the introductory chapter the physical situation of energy transport on molecular aggregates in which the results applies is discussed in detail. This chapter also introduces the nonlinear Schrr odinger model in one and two dimensions , discussing the soliton solutions in one dimension and the collapse phenomenon in two dimensions. Also various analytical methods are described. Then a derivation of the nonlinear Schrr odinger equation is given, based on a Davydov like system described by a tight-binding Hamiltonian and a harmonic lattice coupled by a deformation-type potential. This derivation results in a two-dimensional nonlinear Schrr odinger model, and considering the harmonic lattice to be in thermal contact with a heat bath we show that the nonlinear Schrr odinger equation must be augmented by a multiplicative noise term. The resulting stochastic model is investigated in the continuum limit and the behavior of the coherent excitations under the innuence of the noise is studied. Similarly, we study the uctuation eeects on the two-dimensional collapse phenomenon. We nd numerically and analytically that the collapse can be delayed and ul-timatively arrested by the uctuations. Allowing the system to reach thermal equilibrium we further augment the model by a nonlinear damping term and nd that this prohibits collapse in the strict mathematical …