Investigations On Integrable Properties And Soliton Solutions For Some Nonlinear Models With Symbolic Computation

As a new branch of science, the nonlinear science is an interdiscipline subject in-vestigating nonlinear phenomena of the real world. Since many models and phenomena in the world are nonlinear, the nonlinear science has attracted more and more research interests. As an important branch of the nonlinear science, soliton theory has been paid much attention and some achievements have been arrived at. Actually, the development of soliton theory is really bound up with the nonlinear partial differential equation, while it is difficult and complicated to solve the nonlinear partial differential equation. With the development of the computer technique, symbolic computation has been widely used in many science and engineering fields. By means of symbolic computation, solving the nonlinear partial differential equation becomes comparatively easy. The main content of this dissertation is to investigate the integrable properties, soliton solutions and dynamic features of solitons for some nonlinear models.Main conclusions and contents in this dissertation are listed as follows:(1) Investigations on the reduced Maxwell-Bloch model describing propagation of the ultrashort pulse in optic fibers. We have derived multi-soliton solutions for this model and verified that the interactions of solitons are elastic using the asymptotic analysis method with symbolic computation, and have investigated periodic propagation of solitons by using the figures.(2) Investigations on the generalized homogeneous and inhomogeneous reduced Maxwell-Bloch models. For the homogeneous model, we have discussed the Lax inte-grable property, derived infinite many conservation laws, constructed N-fold Darboux transformation and arrived at multi-soliton solutions, in addition, we have found that the multi-soliton complex will take on when suitable parameters are chosen. For the in-homogeneous model, we have derived the constraint conditions for Painleve integrability and Lax integrability, under these conditions we have derived multi-soliton solutions and found that with suitable parameters the soliton decomposition will take place. (3) Investigations on the generalized inhomogeneous Schrodinger-Maxwell-Bloch model describing propagation of the optic pulse in the two-level medium. With symbolic computation, we have derived the constraint conditions for Painleve integrability and Lax integrability for this model, and have constructed Darboux transformation and N-fold Darboux transformation, derived multi-soliton solutions and analyzed dynamic features in three cases.(4) Investigations on a nonlinear model in fluid mechanics. For the homogeneous model, we have performed modulational instability analysis, and derived the constraint conditions for Painleve integrability and Lax integrability, arrived at multi-soliton solu-tions, and found that bound solitons appear with suitable parameters. For the inhomoge-neous model, we have discussed the Painleve integrable properties, derived multi-soliton solutions, and investigated the dynamic features of solitons.The nonlinear models discussed in this dissertation all have important physical significances and widely practical values in optic fibers and fluid mechanics. The au-thor warmly hopes that the methods and main conclusions in this dissertation can be beneficial to the investigations in the fields of optic fibers, fluid mechanics and applied physics.