Algebro-geometric Quasi-periodic Solutions to the Three-Wave Resonant Interaction Hierarchy

Based on two sets of the Lenard recursion gradients, we derive the three-wave resonant interaction (TWRI) hierarchy associated with a 3× 3 matrix spectral problem. Resorting to the characteristic polynomial of the Lax matrix for the TWRI hierarchy, we introduce a trigonal curve Kₘ₋₂ of arithmetic genus $m-2$ with three infinite points and establish the corresponding Baker--Akhiezer function and meromorphic functions on Kₘ₋₂. The TWRI equations are decomposed into a system of Dubrovin-type ordinary differential equations. Using the theory of the trigonal curve and the properties of the three kinds of Abel differentials, we obtain the explicit theta function representations of the Baker--Akhiezer function, of the meromorphic functions, and, in particular, of solutions for the entire TWRI hierarchy.