Integrable Coupling,Hamilton Structure,Darboux Transformation And Exact Solution For Nonlinear Partial Differential Equations

This dissertation investigates integrable coupling,Hamilton structure,Darboux transformation and exact solution of several kinds of nonlinear partial equations in nonlinear mathematical physics and integrable systems.The main work is carried out in four aspects: a few discrete Lattice systems and their Hamilton structures,Conservation Laws,integrability of nonlinear partial differential equations based on Bell polynomials,integrable coupling and its reduction and the Darboux transformation,exact solution of(2+1)– dimensional integrable system.In Chapter 1,An introduction is devoted to review the research background and the current situation related to this dissertation,including R-matrix theory,several classical methods for seeking exact solutions of nonlinear partial equations and the theory of integrable systems.The main work of this dissertation is also illustrated.In Chapter 2,Using of shift operators and R–Matrix theory,a few discrete Lattice systems and their Hamilton Structures,Conservation Laws are discussed.Applying three shift operators from a shift–operator Lie algebra to generate a few discrete integrable systems with 5-Lattice fields whose Hamilton structures are obtained by making use Poisson tensors of an induced Lie–Poisson bracket.Such integrable systems can be reduced to the well-known Toda Lattice system with a constraint.Next,with the help of Lax representations of the discrete integrable systems,we discover the recursion operators,which can be used to deducing Darboux transformations of the corresponding discrete integrable systems so that exact solutions could be generated.Finally,we apply a reduction of the shift operators presented in the paper to deduce a new discrete Lattice system.Furthermore,we extend the reduced shift operator to obtain an expanding system with three Lattice vector fields,and their Lax pairs,infinite Conservation Laws are obtained.In addition,we specially point out that the method for generating Hamilton structures a simple and efficient way,which adopts the expansion of gradient of Casimir functions not the Casimir functions themselves.In Chapter 3,The Bell polynomials are developed to investigate a variable-coefficient evolution equations and a Generalized KdV equation.In first section,we generalized the KdV equation for non-uniform media with relaxation which has extensive applications in mathematics and physics a more general integrable equation with variable coefficients and further discuss the corresponding bilinear representation,B?cklund transformation,Lax pair andinfinite Conservation Laws by making use of the Bell polynomials.In second section,the integrability of generalized KdV equation is discussed by using Bell polynomials,including bilinear representation,Lax pair,B?cklund transformation and infinite Conservation Law.In Chapter 4,The integrable coupling and its reduction problems are studied based on Tu scheme,zero curvature equation and Lie algebra.The first part,starting from the spectral problem proposed by Geng – Cao,to deduce an integrable hierarchy(called the GC hierarchy)under the frame of zero curvature equations by the Tu scheme,and obtain its new Hamilton structure.Then a 6-dimensional Lie algebra was constructed to obtain a nonlinear integrable model of GC hierarchy,which was reduced to Burgers equation and further reduced to Heat equation.The Hamilton structure of the extended integrable model was obtained from the variational identity.In addition,we construct another 6-dimensional Lie algebra,obtain the second extended integrable model by using Tu scheme,and obtain its Hamilton structure by using trace identity.By comparison,it is pointed out that the extended integrable models of the two hierarchies of GC equations are different.The second part,we introduce a Lie algebra,and defines its corresponding two Loop algebra,Two spectral problems are constructed by means of Loop algebra,and two integrable dynamical systems are derived from their compatibility conditions.Through about such systems,some interesting nonlinear equation are gotten,such as Burgers equation,KdV-MKdV equation,Kuramoto-Sivashinsky equation and a generalized KdV equation.The third part,Based on the idea that Tu Guizhang and Meng Dazhi Under a frame of matrix Lie algebras established a united integrable model of the AKNS hierarchy,the D-AKNS hierarchy,the Levi hierarchy and the TD hierarchy,we introduce two block-matrix Lie algebras to present an isospectral problem,whose compatibility condition gives rise to a type of integrable hierarchy which can be reduced to the Levi hierarchy and the AKNS hierarchy,and so on.In Chapter 5,The reduction,Darboux transformation and exact solutions of hierachy of equations are studied.we start an operator commutator to introduce an isospectral problem from which we deduce a(2+1)-dimensional shallow water wave(SWW)hierarchy and a(2+1)-dimensional Kaup–Newell(KN)hierarchy by making use of the Tu scheme.[18] As reduced consequences,we get a(2+1)-dimensional shallow water wave(SWW)equation and a(2+1)-dimensional KN equation.Furthermore,we investigate two Darboux transformations of the(2+1)-dimensionalSWW equation.In addition,we have known the fact that the well-known KP equation,the mKP equations and the DS equation,and so on all contain the inverse operators in the variable x,however,the(2+1)-dimensional SWW equation and the KN equation are all differential in x and y which are obtained in the paper,as compared,we make use of a reduction case of the self-dual Yang–Mills equations and the spatial spectral problem of the SWW hierarchy to deduce a(2+1)-dimensional heat equation and a(2+1)-dimensional nonlinear generalized SWW equation which contains the terms of inverse operators in x and y.Further consideration on their Darboux transformations is also an interesting thing to us.