Investigations On Analytical Solutions For Some Nonlinear Evolution Equations

As a new interdiscipline subject, the nonlinear science mainly investigate the common rules for various nonlinear phenomena in the real world.In recent years, more research interests have been focused on the nonlinear science. The soliton theory is an important branch of the nonlinear science and has infiltrated into many nonlinear fields, such as the fluid mechanic, plasma physics, optical fiber communications and life sciences. Now the soliton theory has been a relatively complete mathematical physics subject. During the process of investigating on the soliton theory, researchers have found that the development of the soliton theory is based on investigations on the nonlinear evolution equations. In the real world, many physical models can be described by the nonlinear evolution equations, meanwhile many nonlinear evolution equations can exist soliton solutions under suitable conditions. So by virtue of investigations on the nonlinear evolution equations, we can conclude the soliton solutons for physical models in the real world. The main content of this dissertation is to investigate the integrable properties and analytical solutions for some nonlinear mathematical physics models by means of symbolic computations.Main models in this dissertation are:(1) The generalized nonlinear Schrodinger model described the propagation of the femtosecond pulse in the monomode optical fiber: (2) The generalized complex MKdV-Maxwell-Bloch model governs the propagation of optical solitons in a nonlinear light guide doped with two-level resonant atoms.(3) The generalized coupled nonlinear Schrodinger-Maxwell-Bloch model with loss or gain described the propagation of optical solitons in a nonlinear light guide doped with the two-level resonant atoms:(4) The Navier-Stokes model in the nonlinear field:Main investigations on those models are the integrability properties, the Darboux transformation algorithm and the analytical solutions. By virtue of the graphic simulation, the dynamic features of solitons are discussed. We hopes that the methods and main results in this paper may be beneficial to the investigations in the fields of optic fibers, fluid mechanics and applied physics.