Three Generalized Akns Integrable Hierarchies And Some Applications Of The (g′/g)-expansion Method For The Nonlinear Evolution Equations

The major contents in this paper include:three generalized AKNS integrable hierarchies and some applications of the (G'/G)-expansion method for the nonlinear evolution equations.The first chapter, historical origin and researches summarization of soliton theory together with its research meaning are presented.The summary of the major contents, purpose, significance, value and innovation of the research in this paper.The second chapter divided into two parts:In the first part, firstly, based on a well-known Lie algebra G1, consider the corresponding loop algebra, design an isospectral problem, with the help of the the stationary zero curvature equation and the Tri-trace identity, the generalized AKNS hierarchy and its Hamiltonian structure are obtained. Secondly, based on a well-known Lie algebra A1, expand Lie algebra A1 into the Lie algebra G2, construct the corresponding loop algebra, design another isospectral problem, with the help of the the stationary zero curvature equation and the Tri-trace identity, another generalized AKNS hierarchy and its Hamiltonian structure are obtained. Finally, verification that the hierarchy from the Lie algebra G2 can be reduce to the hierarchy from the Lie algebra G1. In the second part, firstly, based on the vector Lie algebra, a new commutator is defined. Secondly, based on the extended trace identity, an extended Tri-trace identity is constructed. Finally, with the help of the stationary zero curvature equation and the Tri-trace identity, a generalized AKNS hierarchy and its Hamiltonian structure are obtained.The third chapter, based on the (G'/G)-expansion method, some traveling wave solutions of (2+1)-dimensional Kp equation and (2+1)-dimensional dispersive Long-wave equation are obtained. The figures of the solutions are presented by Maple program, then analysis the form of solutions and wave fluctuations by the figures of the solutions.