Investigations On Related Problems Of Integrable Systems And Darboux Transformation Of The Soliton Equations

The major contents in this paper include: the integrable system and Darbouxtransformation of the soliton equations. Among them the following two topics aretaken to study the integrable system of the soliton equations: the formulationand integrability of the soliton equations hierarchy and the expanding integrablemodels of integrable hierarchies. In the second chapter, firstly, a type of(2+1)-dimensional multi-component integrable hierarchy and Tu hierarchy arerespectively obtained with the help of the (2+1)-dimensional zero-curvatureequation and Tu scheme and their Hamiltonian structures are solved. Then a newisospectral problem is designed on the base of the Lie algebra A₁. It followsthat a type of new integrable system is obtained by using of Tu scheme. In thethird chapter, we derived the integrable couplings of some integrable hierarchiesby doing further research on the integrable system of the soliton equations.Firstly, we expanded the loop algebras presented in the second chapter for newhigh-dimension loop algebras, designed a new isospectral problem and acquiredan integrable coupling of the (2+1)-dimension Tu hierarchy obtained above.Secondly, a six-dimension Lie algebra is established by the linear combinationof a subalgebra, then its corresponding loop algebra is constructed and theintegrable coupling of the BPT hierarchy is presented by use of the generalizedisospectral problem and Tu sheme. Thirdly, a vector loop algebra and its extendedloop algebra are proposed, which are devoted to obtaining the Tu hierarchy andits integrable coupling. Furthermore, we apply the quadratic-form identity tothe integrable coupling system. Finally, a higher-dimension 6×6 matrix Liealgebra is given, which can be used to directly construct the integrable couplingsof the soliton integrable systems. We obtain the integrable coupling of a newintegrable system and apply the quadratic-form identity to it. In the fourth coupled nonlinear evolution equation is set up with the help of a gauge trans-formation of a spectral problem. As a reduction, a Darboux transformation of thenonlinear schr(o|¨)dinger equation is obtained, furthermore, the exact solutions of theequation are given by applying its Darboux transformation.