On the Riemann Theta Function of a Trigonal Curve and Solutions of the Boussinesq and KP Equations

Abstract. Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable volution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular eal solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve e94 = (2- El) (2- E2) (2- E3) (2- E4). It is shown that the associated Riemann theta function may be d composed into a sum containing two terms, each term being the product of three one-dimensional theta functions. The space and time variables of the KP and Boussinesq equations enter into the arguments of these one-dimensional theta functions in a linear way. 1