Reduction of theta functions and elliptic finite-gap potentials
As it is well-known, special real-valued solutions \(u(x,t)\) of the famous Korteweg-de Vries equation \[ u_t = 6uu_x - u_{xxx} \] can be constructed from hyperelliptic Riemann surfaces of genus \(g\) and their Riemann theta functions (with respect to a period matrix \(B)\) by the formula \[ u(x,t) = - 2 {\partial^2 \over \partial x^2} \ln \theta (Ux + Vt + W,B) + C, \] where \(U,V\) and \(W\) are certain complex \(g\)- vectors and \(C\) is a constant. These solutions appear as quasi-periodic potentials of the Schrödinger operators \[ H(t) = - {\partial^2 \over \partial x^2} + u(x,t) \] which, in turn, define the KdV equation as an operator equation in Lax form. The basic question of reduction theory is the following: Under what conditions, for the chosen hyperelliptic Riemann surface \(\Gamma_g\) and the data \(U,V,W\) and \(B\), is the so- called \(g\)-gap potential \(u(x,t)\) defined above reducible to such ones described in terms of elliptic functions? Generally, this is the problem of reducing Riemann theta functions and Abelian integrals to those in lower genus, especially in genus one, and in this setting it is a classic topic developed by Weierstrass, Poincaré, and others afterwards. In a whole series of previous papers, published between 1982 and 1989, the authors have investigated this problem for \(g\)-gap potentials under various aspects. A rather comprehensive account of their work can be found in their (and other co-authors') recent monograph, there mainly in Chapter 7 [\textit{E. D. Belokolos}, \textit{A. I. Bobenko}, \textit{V. Z. Enol'skij}, \textit{A. R. Its} and \textit{V. B. Matveev}, Algebro-geometric approach to nonlinear integrable equations (1994; Zbl 0809.35001)]. The present paper may be regarded as a (nearly self-contained) continuation of their earlier work, with special emphasis on the reduction of \(g\)-gap potentials to elliptic ones, including the classical Lamé potential and the more recent Treibich-Verdier potentials [\textit{A. Treibich} and \textit{J.-L. Verdier}, Solitons elliptiques (1990; Zbl 0726.14024)]. The exposition is very lucid and detailed, partly a sort of systematic survey on the topic as a whole, however enhanced by several new results in reduction theory and numerous concrete examples.
scientific article; zbMATH DE number 701036
scientific article