Integrable System And Darboux Transformation For Several Kinds Of Equations

Integrable coupling system is a hot content in nonlinear science,which can derive many worthful nonlinear soliton equations.In this paper,we start from a antisymmetric matrix Lie algebra problem.Because most of works focus on 2×2matrix spectral problem,there is less work to search for 4×4 matrix spectral problem.A Darboux transformation(DT)method for solving nonlinear solition equations is a common method.Most people focus on genreal soliton equations,and there is less work to search for complex 3×3 Schr?dinger hierarchies and super-integrable hierarchies.According to the above problems,the dissertation presents main research contents as follows:In the second chapter,we obtain a new loop algebra by using six elements of matrix Lie algebra and expand AKNS hierarchy with 2×2 Lax pairs to AKNS hierarchy with 4×4 Lax pairs,which can get its integrable coupling AKNS system.We construct a 4×4 Lax pairs with antisymmetric matrix Lie algebra.By zero-curvature representation,a recurrence operator L is presented,then we get a new integrable coupling equation hierarchy and find a generalized AKNS equation hierarchy.At last,its Hamiltonian structure is obtained through the trace identity and Tu scheme.In this chapter,we start from a antisymmetric matrix Lie algebra problem.Because most of works focus on(1+1)-dimensional hierarchy,there is less work to search for the(2+1)-dimensional hierarchy.Starting from(2+1)-dimensional hierarchy spectral problem,we get a generalized AKNS equation hierarchy by zero-curvature representation.We choose the appropriate initial values,and(2+1)-dimensional AKNS hierarchy is obtained through Tu scheme.In the third chapter,we first explore Darboux transformation for complex 3×3matrix spectral problem and their exact solutions.Starting from a 3×3 matrix spectral problem,a Darboux transformation method for coupled Schr?dinger equation(CNLSE)is constructed.Then,We mainly around the Lax pairs of super integrable hierarchies.Taking the super-AKNS hierarchy and super-Dirac hierarchy for example,a Darboux transformation(DT)method for two super-integrable hierarchies is constructed.Solition solutions of super-AKNS hierarchy and super-Dirac hierarchy are given by using the Darboux transformation.At last,using maple software,we select the apppropriate parameters,and their figures are plotted.The method is also widely applied to the other super-integrable hierarchies.