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Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions

Shigeki Matsutani
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Abstract

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and \break Kadomtsev-Petviashvili (KP) equations were constructed for a given curve $y^2 = f(x)$ whose genus is three. This study was based upon the fact that about one hundred years ago (Acta Math. (1903) {\bf{27}}, 135-156), H. F. Baker essentially derived KdV hierarchy and KP equation by using bilinear differential operator ${\bold{D}}$, identities of Pfaffians, symmetric functions, hyperelliptic $σ$-function and $\wp$-functions; $\wp_{μν} = -\partial_μ\partial_ν\log σ$ $= - ({\bold{D}}_μ{\bold{D}}_νσσ)/2σ^2$. The connection between his theory and the modern soliton theory was also discussed.

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Shigeki Matsutani. 2001-03-04. Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions. https://doi.org/10.1088/0305-4470%2F34%2F22%2F312

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