Classical And Quantum-mechanic Solitons In The One-dimensional FPU Lattice

Soliton extensively exists in nature, and it is one of the first observed nonlinear phenomena and can be realized in laboratory. With in-depth study of the soliton, the research methods and the applications of the soliton have occupied an important place in various fields of physics. The intrinsic localized modes, firstly found by Sievers and Takeno in study of the FPU model, is an important milestone, which extensively stimulate the research about soliton in lattice. So far, by using the methods of classical mechanics, most of works focus on the classical lattice systems. Generally, the quantum effects are completely neglected in these studies. With the great progress of the nanotechnology on the basis of quantum devices and mesoscopic systems, quantum effect has attracted many attentions and has played a crucial role in many systems. Therefore, the researches about quantum nonlinear excitation become one of the most important research topics. By combining the discrete approximation with the methods of multiple scales and the time-dependent variational principle respectively, this thesis investigates the evolution of the soliton in one-dimensional nonlinear FPU lattice, where the initial conditions are given by the classical method. And some interesting results are clearly demonstrated.The thesis consists of four chapters. In chapter one, we give a brief review to the nonlinear science and mainly introduce the basic concepts about soliton and five typical model of nonlinear lattice. In chapter two, several traditional methods for the study of soliton are introduced. Also, the typical nonlinear equations and corresponding solutions of soliton are represented. In chapter three, we develop a simple version of the method of multiple scale and investigate the soliton in one-dimension classical FPU lattice. Soliton obtained by using this simplied method, is shown to be stable, which confirm this method is effective and efficient. In chapter four, The single-particle state is taken to be a Jackiw-Kerman wave function. by means of the Dirac's time-dependent variational principle with a Hartree-type many body wave function, we study long-time dynamical behaviors of the soliton in 1D quantum FPU chain. Under the condition of minimum uncertainty, equations of motion for the particle expectation values are derived to investigate the stability of soliton. It is shown that the soliton can be stable for a long time in 1D quantum FPU lattice with small effective Plank constant, which is similar to those of the classical model.Finally, we present a conclusion for our work and some prospects for future works in this field.