Hyperelliptic Riemann surfaces of infinite genus and solutions of the KdV equation
type: Item
instance of: scholarly article
title: Hyperelliptic Riemann surfaces of infinite genus and solutions of the KdV equation
zbMATH Open document ID: 0962.35151
DOI: 10.1215/S0012-7094-98-09114-1
author: Werner Müller
author: Robert Schrader
author: M. U. Schmidt
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The authors study hyperelliptic Riemann surfaces of infinite genus associated with infinite-gap solutions to the KdV equation \[ u_t= 6uu_x -u_{xxx},\qquad u|_{t=0}=q \tag{1} \] where \(q\) is a function of \(x\in\mathbb R.\) Their approach is based on an extension of the Its-Matveev formula: Let \({\mathcal M}\) be a hyperelliptic Riemann surface of genus \(g\) (= number of gaps) associated with the fixed gap structure. Let \(\Omega\) be the period matrix of \({\mathcal M},\) and let \(\vartheta_g(z|\Omega)\) be the corresponding Riemann theta function. Then there exist \(U, V\in\mathbb R^g\) and a constant \(C\) such that for every \(W\in\mathbb R^g,\) \[ u(x,t) =-2\frac{d^2}{dx^2}\log\vartheta_g(Ut +Vx + W|\Omega) +C \tag{2} \] is a solution of the KdV equation. The vectors \(U\) and \(V\) are given by periods of certain normalized meromorphic \(1\)-forms on \({\mathcal M}\) with a unique pole at \(\infty\) of order \(2\) and \(4\), respectively, and \(C\) is also determined by \({\mathcal M}.\) The solutions obtained in this way are quasi-periodic. To extend the above formula the authors make use of a Riemann theta function that exists for infinite-genus surfaces whose definition is due to \textit{K. Tahara} [Nagoya Math. J. 33, 57-73 (1968; Zbl 0167.19102)]. The authors consider in the present paper surfaces \({\mathcal M}={\mathcal M}({\Lambda})\) determined by a sequence of ramification points of the form \[ \Lambda :\lambda_0=0<\lambda_1 <\lambda_2 <\cdots\rightarrow \infty \] where they assume that \(\Lambda\) has no finite points of accumulation. Let \(A_1, B_1, A_2, B_2,\ldots \) be a certain canonical homology basis of \({\mathcal M}.\) They construct explicitly a basis \(\phi_1, \phi_2,\ldots \) for the space of holomorphic square integrable \(1\)-forms on \({\mathcal M},\) which is normalized by \[ \int_{A_i}\phi_j =\delta_{ij}. \] Then they prove that such a basis is uniquely determined and, the period matrix of \({\mathcal M}\) is defined by \[ \Omeg…
reviewed by: Ricardo Sa Earp
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37K20
Mathematics Subject Classification ID: 14H55
zbMATH DE Number: 1425080
zbMATH Keywords: infinite-gap solutions
zbMATH Keywords: KdV equation
zbMATH Keywords: Its-Matveev formula
zbMATH Keywords: Riemann theta function
zbMATH Keywords: holomorphic square integrable \(1\)-forms
zbMATH Keywords: Siegel upper half-space
MaRDI profile type: Publication
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full work available at URL: s0012-7094-98-09114-1
OpenAlex ID: W1585573965
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description: scientific article; zbMATH DE number 1425080