Modulational Instability And Discrete Breathers In Nonlinear Lattices

Soliton is a nonlinear phenomenon widely existing in nature,which is widely applied in condensed matter physics,nonlinear optics,plasma physics,biophysics,etc.Solitons are supported in nonlinear lattices because of dispersion action and nonlinear interaction.Lots of reports about nonlinear lattices have been published in the past few decades.And solitons are also deeply expected in storage and transmission of information due to the stable waveform in propagation.In general,modulation instability can predict the parameter region of soliton solutions with different types.Studying the modulation instability of plane waves in nonlinear lattices may help us understand the formation mechanism and physical properties of solitons.We studied the modulational instability of?~4 lattice model and the anisotropic spin lattice model,and the analytical solutions of breather were obtained through the semi-discrete multiscale methodThe thesis consists of five chapters:In chapter one,we mainly introduce the development history and related contents of nonlinear science,and discuss the potential application value of soliton.In chapter two,the oscillator lattices and spin lattices as well as the coherent state method and the semi-discrete multiscale method used in the research process are mainly introduced.In chapter three,we studied the modulational instability analytically and numerically in?~4 lattice model.The result show that the linear stability analysis and numerical simulation are in good agreement within short time scale,but the plane-wave may became unstable in long time scale because of combined wave and third harmonic.We also studied influence of the next-near-neighbor interaction on modulational instability in?~4 lattice model.The results showed that the next-near-neighbor interaction lead to unstable regions increased obviously and move towards the center of brillouin zone.In chapter four,the modulation instability and discrete breathers in the spin lattice model with anisotropy of exchange action are studied.The analytical solutions of discrete breathers in the system are solved by the semi-discrete multiscale method and the types of solitons in different regions are given.The results show that the anisotropy of the next-neighbor interaction and exchange interaction determine the type of breathers.Chapter five is summary and prospect.