Hyperelliptic Kleinian functions and applications

Abstract

We develop the theory of hyperelliptic Kleinian functions. As applications we consider construction of the explicit matrix realization of the hyperelliptic Kummer varieties, differential operators to have the hyperelliptic curve as spectral variety, solution of the KdV equations by Kleinian functions.

Hyperelliptic reduction of the Benney moment equations

“…Thus (4) and (10) have the same characteristics. Any function of λ and f must satisfy the same equation.…”

“…If we now hold p constant, this gives ∂λ ∂t + p ∂λ ∂x − ∂A 0 ∂x ∂λ ∂p = 0 (10) which is a Vlasov equation of the same form as (4). Thus (4) and (10) have the same characteristics.…”

“…Following [4], we define a set of holomorphic differentials du i , and the associated set of second kind differentials dr i . In this genus 2 case these are…”

“…For genus 2, the corresponding partial differential equations were established by Baker [1] (quoted in [4]); these are…”

“…Setting t = ξ −2 and expanding the RHS in terms of this local parameter, ξ, we obtain the series (u 1 −ũ 1 ) = − 1 3 ξ 3 + µ 4 40 ξ 5 + O(ξ 7 ) and (u 2 −ũ 2 ) = − ξ + µ 4 24 ξ 3 + O(ξ 5 ). If we now, for example, take the first equation (31)…”