The Integrable Systems Of Soltons Equtions And Darboux Transformation

The major contents in this paper include:the integrable systems of the soliton equations and Darboux.Among them the following two topics are taken to study the integrable system of the solution equations:the formulation and integrability of the integrable system of the soliton equations hierarchy and the expanding integrable models of integrable hierarchies.In the second chapter,firstly,several(2+1)-dimensional isospectral problems are established based on the loop algebra,(?)₁.As its application,(2+1)-dimensional TB hierarchy is given. Secondly,vector loop algebra(?)_M is constructed and the type of multi-component DLW integrable system is obtained with the help of the(2+1)-dimensional zero-curvature equation and Tu scheme.In third chapter,a matrix isospectral problem is established,then a discrete integrable hierarchy is worked by using dirscrete Tu scheme,which possess Hamiltonian structure.In fourth chapter,firstly,we expand the loop algebra,(?)₁ to the loop algebra(?)₆, design a new isospectral problem and acquired an integrable coupling of the (2+1)-dimensional TB hierarchy.It followed that the Hamiltonian structure of the above system is presented by taking advantage of the extending trace identity—quadratic-form identity.Secondly,the integrable coupling of the multi-component DLW equation hierarchy is given by constructing a higher- dimension loop algebra(?)_M.Finally,a higher- dimension 6×6 matrix lie algebra is given,which can be used to directly construct the integrable couplings of the solution integrable systems.We obtain the integrable coupling of a new integrable system and apply the quadratic-form identity to it.In fifth chapter,a higher- dimension lie algebra is constructed by the method of semi-direct sum of lie algebra and an isospectral problem is designed.As its application,discrete integrable couplings associated with the modified KdV lattice equation are obtained.In the sixth chapter,the explicit N-fold Darboux transformation of a coupled nonlinear evolution equation is set up with the help of a gauge transformation of a spectral problem.By use of Darboux transformation method the single soliton solution and multi- soliton solutions of the soliton equation can be acquired.