Physical effects of nonlinearity in discrete onedimensional systems with spatial variations

Sammanfattning: Physical properties of a solid are affected, to a very high degree, by the spatial structure of the material at hand. This thesis is concerned with such effects in nonlinear systems. In itself, nonlinearity is a subject on the move, and more and more areas of physics tend to take nonlinear effects into account. One direction of nonlinear physics concentrates on coherent structures, e.g., solitary waves, solitons, and breathers. We study how these nonlinear objects behave in onedimensional and discrete systems, dependent on the type of underlying spatial structure of the system.Three types of spatial structures are considered: periodic (including homogeneous) variations; disorder/randomness; and aperiodically ordered systems. The latter of these types may have some properties in common with either of the other two structural types. Sometimes, systems which are aperiodically ordered have inherit properties. A number of such orderings will be considered, including the Fibonacci sequence, the Thue-Morse sequence, and the Rudin-Shapiro sequence. To get a better understanding of the elusive Rudin-Shapiro sequence we construct a new sequence (consisting of the same basic building blocks) which is proven to have a singular continuous electron spectrum in the tight-binding approximation.We investigate how the spectral properties of a system is affected, depending on the underlying ordering of some entities of the system.A nonlinear system that can include spatial variations is an array of Josephson junctions, which are connected in parallel by some kind of inductive couplings. One type of traveling wave that can be found in a Josephson junction array is an approximate kink soliton called a fluxon, i.e., a coherent structure. The fluxon can collide with other fluxons and antifluxons and retain its shape and speed after the interaction.Because of spatial variations in the critical currents or inductive couplings between the functions in an array, a fluxon may be pinned in the array and thus be unable to propagate. This happens for values of a driving current that enable the fluxon to propagate in a homogeneous array. We explain this behavior by using an effective potential for the motion of the fluxon. A fluxon that is moving in a Josephson junction array produces a voltage. A current that is injected into the array, makes the fluxon move, i.e., a relation exists between the current and the voltage (I-V curve) in the array. Depending on the underlying sequence of the critical currents, the I-V curve has distinct properties related to the spatial variation. A new type of dynamical state, which consists of a kink riding on top of a whirling state, is found in a periodic array and a quasiperiodic array. Fluxon bunching is found to be mediated by small amplitude waves also in aperiodically ordered arrays and random arrays.Another type of coherent structure is the discrete breather, which is a time-periodic and spatially localized mode in a discrete system. The system has to be nonlinear so that the frequency of the breather does not resonate with the spectrum for linear (small amplitude) waves of the system. If any such resonances occur, linear waves will be excited. Because the delocalized nature of the linear waves in a periodic system, energy will be carried away from the breather, leading to its destruction. In a discrete system, the linear spectrum is bounded and resonances between the breather frequency or any of its harmonics can be avoided. Discrete breathers seem to be a general feature of systems that are both discrete and nonlinear.We investigate how discrete breathers behave in a Fermi-Pasta-Ulam lattice if two different atoms are aperiodically ordered. The linear spectrum contains a dense set of gaps, which puts the nonresonance condition in a new perspective. We find that discrete breathers exist, and that localized excitations with a chaotic distributions of frequencies can survive longer in aperiodically ordered lattices compared to a periodic lattice.In its natural state, the DNA-molecule is hiding its bases inside the double helix structure. This is done to ensure the best possible protection for the genetic code. In a process of DNA transcription, the base-pairs of the DNA-molecule must be revealed to the surroundings in some way. Preferably, the opening time should be as short as possible. These requirements are met if the DNA-molecule is melted only in the local region, where the transcription is taking place at that moment of time. This local opening of the DNA is called a transcription bubble. As the bubble moves, the DNA is opened in the front and closed at the back, while keeping the number of exposed base-pairs approximately constant.We assume that the transcription bubble has the form of a nonlinear coherent stateWe find that the promoter regions of the bacteriophage T7, where the RNA-coding starts, are more dynamical active compared to the whole genome. The results are checked to be robust by imposing an external disturbance in the form of a thermostat, simulating a constant temperature.