Investigations On Transportation Dynamics Properties Of Solitons For Some Complicated Nonlinear Systems

Nonlinear dynamics is an inter-discipline subject investigating on all kinds of quantitative and qualitative rules in motions especially in evolutionary behaviors for nonlinear systems.Chaos,fractals and solitons are the three main components of study subjects for nonlinear dynamics.Investigations on solitons originate from fluid mechanics,the existences of solitons have been verified from experiments and theories.Since the middle of last century,investigations on soliton phenomena in solid matter have gradually developed,scientists have deduced soliton solutions for some nonlinear evolution equations,and explained the concrete applications in solid matters.Since the early days of this century,scientists have derived nonlinear Schr(?)diner equations(NLSE)in some elastic rods,dispersion rods,round rods and cylindrical shell media,and obtained some valuable results.Based on the above research findings on solitons especially on NLSE in solid matters,this dissertation is to generalize the NLSE from the aspects including adding the complex higher terms,variable coefficients,multi-field coupling effects and discretization of space variables,and to investigate the integrability properties,algorithm for constructing soliton and breather solutions,transportation dynamics properties of solitons and breathers.Concrete research results in this dissertation are listed as follows:1.Investigations on three generalized NLSE with complicated nonlinear and higher dispersion terms,including the mixed NLSE,Hirota model and higher order NLSE delineating the characters of solitons in shallow waters under complex surroundings.We have constructed the iteration algorithm for higher order Darboux matrix corresponding with WKI type linear matrix spectral problems,analyzed concrete generative mechanisms for breathers and bound solitons,discussed the effects of complicated parameters on solitons and breathers.Via the asymptotic analysis method,we have theoretically proved the mechanism of elastic interactions.2.Investigations on the reduced Maxwell-Bloch model with variable coefficients describing the solitons in two level er-doped non-autonomous medium.We have analyzed the Painlev?e properties,obtained the constraint conditions between those variable coefficients,deduced infinite conservation laws,and constructed the Darboux matrix with variable coefficients,calculated out the non-autonomous multi-soliton solutions.3.Investigations on three types of higher order coupled NLSE with complicated nonlinear and higher dispersion terms.We have constructed the Darboux matrix with higher order matrix,derived multi-soliton and breather solutions,analyzed the transformation conditions between solitons and breathers,obtained the periodic solutions and discussed the effects of coherent coupling on solitons and breathers.4.Investigations on the discrete NLSE with variable coefficients describing the energy transmission in ? helical protein.We have constructed the discrete Darboux matrix algorithm,derived discrete soliton solutions.The generalized NLSE models discussed in this dissertation all have important physical significances and widely practical values in fluid mechanics,optic fibers and biology.The author warmly hopes that the methods and main conclusions in this dissertation can be beneficial to the investigations on the nonlinear evolution equations in the fields of solid mechanics and other fields of sciences.