对称约束下的KP型方程簇的可积性

KP方程簇是指基于Sato 理论建立的可积系统，并以著名的KdV方程及其在2+1维空间的推广的KP方程为典型代表的方程簇。KP型方程簇是KP方程簇的推广以及约化，主要包含KP、修正KP、BKP\CKP、离散KP、超KP、q-KP、Matrix-KP、Frobenius-KP等等。基于Sato理论，本项目拟从哈密顿结构与可解性两方面来研究对称约束下的KP型方程簇的可积性：.（A）对称约束下的KP型方程簇哈密顿结构的代数性质，.（B）通过对称约束求2+1维可积方程的精确解，以及对称约束所得到耦合方程的可解性。

The KP hierarchy is an integrable hierarchy established on the Sato theory, which is represented by the KP equation—a two-dimensional extension of the well-known KdV equation. KP type hierarchy means the KP hierarchy and its reductions and extensions, which includes KP, modified KP, BKP\CKP, discrete KP, Super KP, q-KP, Matrix-KP and Frobenius-KP, etc. The aim of this project is to study the integrability of the constrained KP hierarchy, based on the Sato theory, from two aspects of Hamiltonian structure and solvability:.(A) The algebraic property of the Hamiltonian structure for the KP type hierarchy. .(B) Solving (2+1)-dimensional integrable equations by the symmetry constraint, and exploring the solvability of the coupled equations obtained by the symmetry constraint.