Solitory Solutions For Some Schr(?)dinger Equations And Their Stability

Since Bertrand Russell discovered the phenomenon of solitary wave in1834, the phenomenon of solitary wave has been discovered in many fields of science and technology, such as: Hydromechanics, Bose-Einstein condensation, Plasma physics, molecular biology, etc. In the1980s, the conception of optical soliton was introduced. Due to the capability of carrying a large amount of information, high transmission speed, long transmission distance, and strong confidentiality, the optical soliton has been paid great attention, and optical soliton has been widely applied in the optical communication, optical switch, optical driving, optical storage, etc.Up to now, many methods and techniques to explore soliton solutions have been put forward, and many types of soliton are derived. Due to the instability, maybe some solitons cannot be observed in the experiments, thus it cannot be applied. Moreover, Modulation instability has been paid much attention because it can be considered as a precursor of formation to soliton. Therefore, the research on soliton solutions and the behaviors of soliton solutions is valuable in the theory and the application.Firstly, the exact solutions with more arbitrary parameters of2-dimensional Ablowitz-Ladik equation are derived by G’/G-expansion method. The effect of parameters in the2-dimensional Ablowitz-Ladik equation on the stability is analysed using perturbation method. The stable regions of exact solutions are obtained and the variation of the stable region is discussed.Secondly, the modulation instability of2-dimensional Ablowitz-Ladik is analysed. By using analytic and numerical method, the dispersion relationship of the2-dimensional Ablowitz-Ladik is derived and the relationship among the parameters are obtained when modulation instability occurs.Lastly, onsite, inter-site and asymmetric variational solutions of the non-integrable coupled discrete Schr dinger equations are derived and their existence regions are obtained via variational approach and numerical method. The bifurcation line of variational solutions is found. The stability of fundamental solution is discussed and the stable regions are obtained. The evolution of solutions with time is analysed.The result of this paper can provide theoretical support for the study of the optical soliton in waveguide array and the result also can explain many phenomena in regions of Bose-Einstein condensation, Plasma physics, etc.