Study Of Quantum Solitons In One-dimensional Nonlinear Lattice Models

Soliton is a typical and important nonlinear phenomenon. The theory of solitons has important application in many fields ranging from condensed matter physics and plasma physics to biophysics, optical fibers and geological systems. In 1988, by using the rotating wave approximation and Green function method, Sievers and Takeno firstly showed that there exist intrinsic localized modes in the perfect lattices resulting from the interplay between discreteness and nonlinearity. Since then, a vast amount of studies has been poured to nonlinear localized modes in discrete nonlinear lattices. Up to now, most attentions have been paid to classical lattices. However, as we all know, dynamic behaviors are governed by quantum mechanics in microcosmic systems. Therefore, the investigation of solitons in quantum lattices is necessary. Generally, the quantum nonlinear excitations are studied by both numerical and analytical methods. For example, adopting"pseudofield"operators, V. V. Konotop and S. Takeno investigated the quantum one-dimensional FPU chain and had found quantum envelope soliton solution. In addition, by means of Glauber's coherent method combined with continuum approximation, Zhu-Pei Shi et al. investigated the quantum one-dimensional monatomic chain and obtained the nonlinear phonon localized solutions for the expectation value of the displacement. Moreover, by using exact numerical diaganonalization of the Hamiltonian, the multi-phonon bound states (or biphonons bound states) exist as eigenstates, which are counterparts of nonlinear localized modes of classical nonlinear microscopic systems. These results not only give us several interesting informations of the anharmonic lattice, but also imply that nonlinear localized modes can exist in quantum anharmonic lattice. In this thesis, by using of coherent method and time-dependent variational principle, we investigate the quantum Klein-Gordon and quantum FPU chain, some significant results are presented. These results not only enrich the soliton theory in the 1D system but also provide some theoretical basis for other theoretical and experimental works of relevancy.The thesis consists of four chapters. In chapter one, we briefly introduce the basic theoretical concept of nonlinear science, soliton, the history of the study of solitons, three nonlinear equations with soliton solutions, present status and significance of the study of solitons. Chapter two focus on soliton in nonlinear atomic lattices, including brief introduction of lattice models and present status of the study of soliton in classical and quantum atomic lattices. In chapter three, by means of the Glauber's coherent state method combined with multiple-scale method, we investigate the localized modes in a quantum one-dimensional Klein-Gordon chain and find that the equation of motion of annihilation operator is reduced to the discrete nonlinear schr?dinger equation. The results indicate that the model can support both bright and dark small amplitude traveling and non-traveling nonlinear localized modes in different parameter spaces. In chapter four, Dirac's time-dependent variational principle and multi-scale method are used for dealing with quantum FPU chain. Small amplitude nonlinear localized modes are obtained. Finally, we summarize the present thesis and give an outlook of the future study in this field.