Riemann theta function solutions to the coupled long wave–short wave resonance equations

Abstract

Based on the Lenard recursion equations, we derive the Lax pair for the hierarchy of coupled long wave–short wave resonance equations, in which the first nontrivial member is the coupled long wave–short wave resonance equations. Resorting to the characteristic polynomial of Lax matrix for the hierarchy of coupled long wave–short wave resonance equations, we introduce a trigonal curve and a basis of holomorphic differentials on it, from which we construct the Riemann theta function of the trigonal curve, the related Baker–Akhiezer function, and an algebraic function carrying the data of the divisor. By comparing the asymptotic expansions for the Baker–Akhiezer function and its Riemann theta function representation, we obtain the explicit Riemann theta function solutions for the entire hierarchy of coupled long wave–short wave resonance equations.