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We develop the theory of generalized Weierstrass σ-and ℘-functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the ℘-functions, a proof that the coefficients of the power series expansion of the σ-function are polynomials of coefficients of the defining(More)
We consider a hierarchy of the natural type Hamiltonian systems of n degrees of freedom with polynomial potentials separable in general ellipsoidal and general paraboloidal coordinates. We give a Lax representation in terms of 2 × 2 matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the(More)
We use the bilinear operator formalism to derive a new representation of the power series for the Weierstrass σ function. It is not generally known that Baker solved a number of nonlinear integrable partial differential equations in 1907 [2]. In the course of an investigation of ultra-elliptic functions, he wrote certain relations between them, in which(More)
We discuss the theory of generalized Weierstrass σ and ℘ functions defined on a trigonal curve of genus four, following earlier work on the genus three case. The specific example of the " purely trigonal " (or " cyclic trigonal ") curve y 3 = x 5 + λ 4 x 4 + λ 3 x 3 + λ 2 x 2 + λ 1 x + λ 0 is discussed in detail, including a list of some of the associated(More)