On The Integrability Of Several B-type Kdv Hierarchies

One core problem of the soliton theory is the integrability of nonlinear differential equations.There isn't uniform definition about integrability of nonlinear differential equations,in what sense people usually illustrate that the nonlinear differential equations is integrable.such as:Liouville integrability,Lax integrability,inverse scattering integrability,bilinear integrability,symmetric integrable etc.In this paper,we study three B-types of integrable hierarchies,their Lax integrability and Liouville integrability have been demonstrated.In the first chapter,we introduce background of the soliton theory and its development status,give the basic definition of integrability and introduce the solving methods of the recursion operation.we present the concept of the conservation laws at last.In the second chapter,we discussed sp(2,C)-counterpart of the hierarchy.Based on 4 x 4 matrix spectral problem,we obtain the recursion operator and integrable hierarchies using by zero curvature equations,and construct bi-Hamiltonian structure by the trace identity.Then we apply block matrix to sp(2,C)-counter part of the hierarchy,we work out infinitely many conservation laws and its recursion operator.In the third chapter,we discussed sl(4,C)-counterpart of the hierarchy.Based on 4 x 4 matrix spectral problem,we obtain the recursion operator and integrable hierarchies using by zero curvature equations,and construct bi-Hamiltonian structure by the trace identity.Then we apply block matrix to sl(4,C)-counterpart of the hierarchy,we work out infinitely many conservation laws and its recursion operator.In the fourth chapter,we discussed so(5,C)-counterpart of the hierarchy.Based on 5 x 5 matrix spectral problem,we obtain the recursion operator and integrable hierarchies using by zero curvature equations,and construct bi-Hamiltonian structure by the trace identity.Finally,we obtain the conservation laws of so(5,C)-counterpart of the hierarchy by using direct method.At last,we draw the conclusion that this three B-types of integrable hierarchies are Lax integrable and Liouville integrable.