标题： Frobenius流形和Frobenius代数值可积系统 显示英文标题

标题： Frobenius manifolds and Frobenius algebra-valued integrable systems

摘要： Frobenius代数值的双哈密顿演化方程，包括无色散和色散的情况，都被研究了。 对于这些方程，依赖场取值于一个固定的Frobenius代数$\mathcal{A}\,.$。在无色散情况下，这些方程是通过研究任意解析的Frobenius流形$\mathcal{M}$与固定代数$\mathcal{A}$的张量积得到的，其中将固定代数视为平凡的Frobenius流形。 得到的主要层次可以表示为$\mathcal{A}$-值场的形式，并且使用相对于$\mathcal{A}$-值场的泛函导数定义双哈密顿结构。 在色散情况下，方程是用$\mathcal{A}$-值Lax方程来定义的，并给出了$\mathcal{A}$-值KP和Toda格子层次的构造。 到一个$\mathcal{A}$值的$\mbox{GD}_m$层次结构和一个扩展的$(M,N)$双分次Toda格子层次结构的约化被给出，并通过采取适当的无色散极限实现了与论文第一部分的关系。 显示英文摘要

摘要： Frobenius algebra-valued bi-Hamiltonian evolution equations, both dispersionless and dispersive, are studied. For such equations, the dependent fields take values in a fixed Frobenius algebra $\mathcal{A}\,.$ In the dispersionless case, these equations are obtained by studying the tensor product of an arbitrary analytic Frobenius manifold $\mathcal{M}$ with the fixed algebra $\mathcal{A}$, considered as a trivial Frobenius manifold. The resulting principal hierarchy can be rewritten in terms of $\mathcal{A}$-valued fields, and the bi-Hamiltonian structure defined using a functional derivative with respect to an $\mathcal{A}$-valued field. In the dispersive case, the equations are defined in terms of $\mathcal{A}$-valued Lax equations, and the construction is given for $\mathcal{A}$-valued KP and Toda lattice hierarchies. Reductions to an $\mathcal{A}$-valued $\mbox{GD}_m$ hierarchy and an extended $(M,N)$-bigraded Toda lattice hierarchy are given, and the relationship to the first part of the paper is achieved by taking the appropriate dispersionless limits.