Study Of Gap Solitons And Their Dynamics In Disordered Systems

In last decades, the interplay between nonlinearity and periodicity has received intensively experimental and theoretical studies. One of the most interesting results is the existence of solitary wave in gap, termed "gap soliton" by Chen and Mills. Due to its special transmission properties, gap soliton can be used for a wide range of applications in optical communications and optical storage. Recently, because it provides to other related fields, such as the rapidly developing field of "Bose-Einstein condensates", the research of gap soliton has received considerable attentions.In this dissertation, firstly, we develop a local-Bloch wave picture to investigate the gap soliton solutions in one-dimensional nonlinear photonic crystal systems. Then we extend the concept of "gap soliton" into the electronic system to study the polaronic soliton solutions in a narrow gap system. Finally, we study the dynamics of matter wave gap soliton in optical lattice with randomness.In chapter 2, we study the gap soliton solutions in one-dimensional nonlinear photonic crystal (periodic Kerr media). First, we introduce two methods: coupled-mode method and multiple-scales method to obtain the gap soliton solutions. To overcome the shortcoming of the two methods, we introduce the local Bloch wave picture. Based on the local Bloch wave picture, we find that the envelop function of the field is a generalized nonlinear Schrodinger equations. Using the envelop equations, we find that the different GS functional forms are from the competition of one linear term and two nonlinear terms. And finally, we give out a simple soliton stable analysis based on the local Bloch wave picture.In chapter 3, we construct a model to investigate the one-dimensional localized excitation, which is called "polaronic soliton" in a narrow gap system. From this model, we can obtain the soliton solution analytically in the continuum limit. We find that the soliton properties strongly depend on the gap regions where the electronic energy falls: for example, the soliton has a bell-like form when the eigenenergy is near the conduction band-edge, but a two-peak form when the eigenenergy is near the gap center. We give out a physical interpretation based on the local energy competition between the electron-lattice attractive energy and the lattice elastic energy. Compare our model with the molecular-crystal model and the coupled electron-phonon model, we find that our model can be treated as a bridge between them: the molecular-crystal model can be seen as a special case of our model and our model can be treated as a spacial case of the coupled electron-phonon model in some limit. Finally, we compare the polaronic soliton in our model and the gap soliton in one-dimensional nonlinear photonic crystal and find the nonlinearity is very important in soliton solutions.In chapter 4, using the numerical finite-difference time-domain method, we investigate the matter-wave gap solitons propagation in random optical lattice. For the weak random case, we introduce the effective particle picture to solve the movement of gap soliton. Based on the effective particle picture, obtain the effect of the randomness on gap soliton and solve out the motions. Moreover, we obtain the general law of the gap soliton movement depending on the weak randomness. Such as, with the increase of the random strength, the ensemble-average velocity reduces slowly in which the reduction is proportional to the variance of the randomness, and the reflected probability increase. For the large random case, we find that the gap solitons can be trapped by the randomness. From the time evolution of the soliton field, we give out a qualitative interpretation: the gap soliton field can be radiated by the large randomness and trapped by two defect modes. And we also simulate the case that gap soliton propagation through two defect to give an interpretation. This may be provide a route of optical memory.