Expanding Integrable Models and Their Some Reductions as Well as Darboux Transformations

DOI：10.3770/j.issn:2095-2651.2016.03.007

 作者 单位 冯滨鲁 潍坊学院数学与信息科学学院, 山东 潍坊 261061 韩耀琮 香港城市大学数学系, 中国 香港

本文首先给出了一个三维李代数H并将它扩展成一个六维李代数$T$,其相应的圈代数分别记为$\tilde H$ 和$\tilde T$.利用圈代数$\tilde H$和屠格式导出了一个可积方程族和它的一个新的达布变换用来求精确周期解.利用圈代数$\tilde T$导出了一个新的可积耦合族并将它约化为一变系数非线性方程,再利用变分恒等式导出了该方程族的哈密尔顿结构.进一步地,我们构造了李代数$H$的一个高维圈代数$\tilde H$导出了含有五个位势函数的新的刘维尔可积方程族,它可约化为一个复的修正KdV方程,利用迹恒等式我们导出了这族方程的三哈密尔顿结构,这是一类导出多哈密尔顿结构的新方法.最后,我们推广了圈代数$\tilde H$导出了一个变系数可积方程族.

In this paper we first present a 3-dimensional Lie algebra $H$ and enlarge it into a 6-dimensional Lie algebra $T$ with corresponding loop algebras $\tilde H$ and $\tilde T$, respectively. By using the loop algebra $\tilde H$ and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra $\tilde T$, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebra $\bar H$ of the Lie algebra $H$ from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex mKdV equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multi-Hamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra $\tilde H$ to obtain an integrable hierarchy with variable coefficients.