链KP型方程族的若干研究

In this program, we investigate the construction of the lattice KP-type hierarchy, as well as the corresponding exact solutions and integrablity. (1) Generalized Cauchy matrix approach will be used to construct lattice KP-type hierarchy, including discrete KP hierarchy, lattice potential KP hierarchy, lattice potential mKP hierarchy, lattice Schwarzian KP hierarchy and KP-type NQC hierarchy. For these discrete integrable hierarchies in multidimension, bilinear method, Cauchy matrix approach, generalized Cauchy matrix approach and Darboux transform will be adopted to seek their several kinds of exact solutions. Meanwhile, dynamical properties for these solutions will also be discussed. (2) Symmetries and conservation laws to the lattice KP-type hierarchy will be constructed. There are strong physical backgrounds for the problems in this program. The fruitful results will develop some corresponding theories in discrete integrable systems.

本项目主要研究链KP型方程族的构造及其精确求解和可积特征。(1)利用广义Cauchy矩阵方法构造链KP型方程族，包括离散KP方程族、链势KP方程族、链势mKP方程族、链Schwarzian KP方程族和KP型的NQC方程族。对于这类高维离散可积方程族，我们拟采用双线性方法、Cauchy矩阵方法、广义Cauchy矩阵方法和Darboux变换等方法构造其多种形式的精确解，并分析解的动力学特征。(2)构造链KP型方程族的对称和守恒律。本项目考虑的问题具有深刻的物理背景，研究成果将在一定程度上丰富离散可积系统的相关理论。

利用广义Cauchy矩阵方法, 我们分别构造了链KP方程族、非自治链KP方程族、扩展链Gelfand-Dikii型方程族、链AKNS方程、链负阶AKNS方程、链负阶势KdV方程、非自治ABS型链方程及非自治扩展链Boussinesq型方程等离散可积系统。基于该方法，我们获得了这些离散可积系统多种形式的精确解，包括孤子解、Jordan块解和混合解。针对非自治链KP型方程，我们讨论了这类方程的Lax表示并研究了多维相容性。借助于链KP方程族的有理解，利用约化我们获得了链势KdV方程和两个半离散链势KdV方程的有理解,并研究了这些解的动力学特征。利用双线性方法，我们研究了非自治ABS链方程中的非自治H1方程、非自治H2方程、非自治H3方程和非自治Q1方程的双线性方程以及Casoratian解。