Investigations On Related Problems Of Integrable Hierarchy

The major contents in this paper include: the formulation of continuous integrable hierarchies, the formulation of discrete integrable hierarchies and the expanding integrable models. In the second chapter, firstly two multi-component matrix loop algebras are constructed based on a basis of loopalgebra (A|-)₁ and several new isospectral problems are designed. Using Tu scheme,the multi-component Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, a type of new multi-component integrable hierarchy and the multi-component Kaup-Newell(KN) hierarchy with its Hamiltonian structure are obtained. Secondly, a new loopalgebra (G|⁾_M with a simple commutative operation just like that in the loopalgebra (A|-)₁ is constructed, which is devoted to establishing a new isospectralproblem. As its application, a multi-component integrable system similar to the NLS-mKdV hierarchy is worked out. In the third chapter, an algebraic systemX is constructed by using the known loop algebra (G|⁾₃. Then a new isospectralproblem is established by taking advantage of X , which is devoted to working out the generalized Volterra lattice hierarchy. The expanding integrable models are studied in the fourth chapter, firstly based on the extended loop algebras of loop algebras in the second chapter, some expanding integrable models are obtained. As its application, the expanding integrable model of AKNS hierarchyis obtained. And an extended algebraic system X of X is presented, from whichthe integrable coupling system of the Volterra lattice is obtained. Secondly, two Lie algebras with 6 dimensions based on the line combination of the Liealgebras ,A₁₁, and A₁₂ are constructed, then their corresponding loop algebrasare obtained and proposed a direct method for the expanding integrable models. As their applications, the expanding integrable models of the multi-component dispersive long wave(DLW) hierarchy and S-mKdV hierarchy are obtained. Last based on the constructed two loop algebras, the two resulting isospectral problems are established. Then using Tu scheme, two types of various expanding integrable models of generalized NLS equation are obtained. Another direct method for the expanding integrable model is proposed.