Algebro-Geometric Solutions of the Coupled Chaffee-Infante Reaction Diffusion Hierarchy

Abstract

The coupled Chaffee-Infante reaction diffusion (CCIRD) hierarchy associated with a $3\times 3$ matrix spectral problem is derived by using two sets of the Lenard recursion gradients. Based on the characteristic polynomial of the Lax matrix for the CCIRD hierarchy, we introduce a trigonal curve ${\mathcal{K}}_{m-2}$ of arithmetic genus $m-2$ , from which the corresponding Baker-Akhiezer function and meromorphic functions on ${\mathcal{K}}_{m-2}$ are constructed. Then, the CCIRD equations are decomposed into Dubrovin-type ordinary differential equations. Furthermore, the theory of the trigonal curve and the properties of the three kinds of Abel differentials are applied to obtain the explicit theta function representations of the Baker-Akhiezer function and the meromorphic functions. In particular, algebro-geometric solutions for the entire CCIRD hierarchy are obtained.