@PHDTHESIS\{IMM1997-02442,
    author       = "K. {\O}. Rasmussen",
    title        = "Nonlinear and stochastic dynamics of coherent structures",
    year         = "1997",
    school       = "Informatics and Mathematical Modelling, Technical University of Denmark, {DTU}",
    address      = "Richard Petersens Plads, Building 321, {DK-}2800 Kgs. Lyngby",
    type         = "",
    note         = "Supervisor: Peter Leth Christiansen",
    url          = "http://www2.compute.dtu.dk/pubdb/pubs/2442-full.html",
    abstract     = "This Thesis deals with nonlinear and stochastic dynamics in systems which can be described by nonlinear Schr{\"{o}}dinger models. Basically three different models are investigated. The first is the continuum nonlinear Schr{\"{o}}ndinger model in one and two dimensions generalized by a tunable degree of nonlinearity. Various perturbations such as fluctuating forces, localized impuri ties, and dissipation are considered. The second kind of model is the discrete nonlinear Schr{\"{o}}ding er model, which also is investigated in one and two dimensions. External perturbati ons similar to the continuum case are also considered in the discrete model. Finally, nonlinear Schr{\"{o}}dinger continuum and discrete models with nonlocal d ispersion 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 Schr{\"{o}}dinger 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 Schr{\"{o}}dinger equation is given, based on a Davydov like system described by a tight-binding Hamiltonian and a harmonic lattice coupled b y a deformation-type potential. This derivation results in a two-dimensional nonline ar Schr{\"{o}}dinger model, and considering the harmonic lattice to be in thermal contact with a heat bath w e show that the nonlinear Schr{\"{o}}dinger 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 influence of the noise is studied . Similarly, we study the fluctuation effects on the two-dimensional collapse phenomenon. We find numerically and analytically that the collapse can be delayed and ultimatively arrested by the fluctuations. Allowing the system to reach thermal equilibrium we further augment the model by a nonlineardamping term and find that this prohibits collapse in the strict mathematical se nse. However a collapse like behavior still persists in the presence of the nonlinear damping . Apart from the absence of the collapse in the strict mathematical sense we find that the nonlinear damping term has rather weak influence on the interplay between fluctuations and self-focusing. The study of the continuum mod el is concluded by an investigation of the dynamics of localized states in the vici nity of an impurity.

Studying the discrete nonlinear Schr{\"{o}}dinger model, we first analyze the intrinsically localized excitations supported by this model in one dimension . This analysis is accomplished using analytical methods developed for nonlinear maps. It is demonstrated how the nonanalyticity of the map through homoclinic an d heteroclinic connections permits the existence of localized states on the lattice. The pinnin g effect of the discrete lattice is also investigated, constructing a Melnikov function describing qualit atively the difference between on-site and inter-site states. Since the intrinsically localized excitations are rather robust we further study the implications of fluctuations and nonlinear damping in this discrete model. The f luctuations are found always to destroy the localized states.

Existence and dynamics of the intrinsically localized excitations in the two-dimensional discr ete model are also studied. It is found that in two dimensions a bistability phenomenon of the localized states appears. The bistability expresses itself by allowing localized states of various width to have equal norms. We find in the two-dimensional discrete model that the interplay of the collapse effect and the discrete pinning allows dynamical creation of a spatially distributed se t of localized states from a broad initial excitation.

The last kind of models studied in the Thesis is nonlinear Schr{\"{o}}dinger models with nonlocal dispersive inter action. First a continuum model with an exponential dependence of dispersive interaction is stud ied. This model shows in contrast to the ordinary continuum nonlinear Schr{\"{o}}dinger models that the nonlocality imposes an upper bound on the norm of a possible localized excitation. The model is also shown to suppo rt a cusp soliton. A similar discrete nonlocal model is discussed. This model has an algebraic dependence of the dispersive interaction. There exists no upper limit of the norm in the discrete model, but the possibility of a bistability phenomenon similar to that of the two-dimensional model is shown to occur.

Finally, we show that a two-dimensional Kronig-Penney model describing for examp le propagation of electromagnetic waves in photonic bandgap materials can be reduced to a one-dimensional nonlocal nonli near Schr{\"{o}}dinger model, which is similar to the nonlocal models considered previously."
}