On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations

Recently, considerable progress has been made in understanding the nature of the algebro-geometrical superposition principles for the solutions of nonlinear completely integrable evolution equations, and mainly for the equations related to hyperelliptic Riemann surfaces. Here we find such a superposition formula for particular real solutions of the KP and Boussinesq equations related to the nonhyperelliptic curve ⁴ = ( – E ₁) ( – E ₂) ( – E ₃) ( – E ₄). It is shown that the associated Riemann theta function may be decomposed 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.On leave from Leningrad State University and Leningrad Institute of Aviation Instrumentation.