Study Of Two-vibron Bound States In One-dimensional Nonlinear Lattices

Soliton is a universal nonlinear phenomenon, which exists in nature ranging from macrophysics to microphysics. As a microscopic model, study of solitons in the nonlinear atomic lattice model has proven to be a hot topic in this field till now. Especially, since 1988 when Sievers and Takeno discovered intrinsic localized modes (also referred to as discrete breathers) resulted from interaction between discreetness and nonlinearity, a vast amount of researches have been orderly done. In some 1D discrete lattice systems, quantum effects play a crucial role in determining the properties of systems, and should be taken into account. As a consequence, looking for the quantum-mechanic counterpart of the discrete breathers in lattice has gradually become a hot topic in this field. Recently, with the help of correlation functions, the two-vibron bound states in the nonlinear quantum mechanic lattice models show distinctly the particle properties through the study of Bose-Hubbard, Klein-Gordon models and so on. So far, such two-vibron bound states are acknowledged to be the simplest quantum breathers. Although the 1D quantum nonlinear lattice is formally simple, it is still a many body quantum-mechanic problem with quite complicated dynamics. Consequently, one has to adopt some approximation to solve such problems. In this thesis, we investigate the eigenvalue problem of the quantum Fermi-Pasta-Ulam-βmodel by adopting some approximation in order to search the quantum-mechanic counterpart of the discrete breathers, which is expected to play an important role in understanding properties, such as energy transport, of various low dimensional materials.The thesis consists of four chapters. Chapter one gives a brief introduction of the nonlinear science and the development of solitons. Several typical nonlinear equations which have soliton solutions are also presented. In chapter two, we give some description of the nonlinear lattice model and the status quo of soliton researches in nonlinear lattice models.Chapter three is devoted to researching status of vibron bound states in the one dimensional quantum nonlinear lattice models. The most of length of this chapter is devoted to the two-vibron bound states in the Bose-Hubbard and Klein-Gordon nonlinear lattice models.In chapter four, we first quantize the Fermi-Pasta-Ulam-βmodel within the number conserving approximation. Then, we study the two-vibron bound states of the system in the presentation of the number states. The results indicate that it is the nonlinearity that leads to the appearance of discrete band in the energy spectrum. We found that energy gap at the center of Brillouin zone is larger than at the edge of Brillouin zone. And the analysis of the discrete band show that three different types of the two vibron bound states may exist in the system. Moreover, the phenomenon that the wave number has a significant effect on such quantum localized states is also found.Finally, we present a conclusion for our work and some prospects for future works in this field.