New elliptic potentials
The well-known method of finite-gap integration, based on the Krichever dictionary between certain data on Riemann surfaces and algebras of commuting differential operators, allows to construct quasi-periodic solutions of particular classes of nonlinear integrable differential equations in terms of theta functions of Riemann surfaces. One of the most interesting questions, in this framework, is how to find as many as possible solutions of integrable equations in terms of elliptic functions. A great progress, in this direction, was the joint work of J.- L. Verdier and the author of the present paper, the first account of which appeared in 1987 [cf. \textit{J.-L. Verdier}, `New elliptic solitons', in: Algebraic Analysis, dedicated to M. Sato on the occasion of his 60th birthday, Vol. 2, 901-910 (1989; Zbl 0692.35090)]. The author and Verdier succeeded in constructing new elliptic solutions (finite-gap potentials) for the Korteweg-de Vries equation (KdV) and Kadomtsev-Petviashvili equation (KP), which came rather unexpected and revived the interest in this area. Their method of using the so-called tangential covers of elliptic curves (cf. reference above), for which they elaborated an algorithmic procedure, turned out to be extremely effective and led to the discovery of a lot of new elliptic finite-gap potentials of degrees 3, 4, 5 and 6. After the sudden tragic death of J.-L. Verdier, the author of the present paper kept working on this fruitful approach, and the article under review is a report on the present state of the Treibich-Verdier approach to constructing explicitly elliptic finite-gap solutions for the KdV and KP equations. As such, it represents a self-containing and comprehensive, nevertheless down-to earth survey, giving complete proofs in the case of smooth curves [cf. the author, Duke Math. J. 59, No. 3, 611-627 (1989; Zbl 0698.14029); cf. also the author and \textit{J.-L. Verdier}, ``Solitons elliptiques'' in: The Grothendieck Festschrift, Prog. Math. 88, 437-480 (1990; Zbl 0726.14024)], and explaining their so far obtained generalizations [cf. the author and \textit{J.-L. Verdier}, C. R. Acad. Sci. Paris, Sér. I, 311, No. 1, 51-54 (1990; Zbl 0712.33014)]. The exposition is written in a very careful, lucid and inspiring style. It provides a useful and systematic introduction to, and a beautiful overview of the powerful Treibich-Verdier method of integrating soliton equations.
scientific article; zbMATH DE number 701033