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On the Riemann theta function of a trigonal curve and solutions of the Boussinesq and KP equations. (English) Zbl 0641.35061
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 \(\omega^ 4=(\lambda -E_ 1)(\lambda -E_ 2)(\lambda -E_ 3)(\lambda -E_ 4)\). 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.

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
35A30 Geometric theory, characteristics, transformations in context of PDEs
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