Hamiltonian Structures For Frobenius Algebra-valued Constrained KP Hierarchies

The KP hierarchy and its generalizations are an important research topic in the clas-sical integrable theory. In [70,74], Strachan and Zuo introduced an Frobenius-algebra valued KP hierarchy, which is a commutative generalization of the KP hierachy. This thesis is a continuation to the work in [70,74]. Firstly, we will recall the Frobenius-algebra valued KP hierarchy and its Hamiltonian structure; Afterwards, we will in-troduce two kinds of constrained hierarchies:one is the Frobenius-algebra valued n-constrained KP hierarchy; the other is the Frobenius-algebra valued (n, m)-constrained KP hierarchy; Finally, we will construct their Hamiltonian structures and study their related properties, such as the Kupershmidt-Wilson-type theorem.