On Algebro-Geometric Simply-Periodic Solutions of the KdV Hierarchy

Abstract

In this paper, we show that as $$\tau \rightarrow \sqrt{-1}\infty $$, any zero of the Lame function converges to either $$\infty $$ or a finite point p satisfying $${\text {Re}}p=\frac{1}{2}$$ and $$e^{2\pi i p}$$ being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line $${\text {Re}}z=\frac{1}{2}$$. We also compute explicitly the eigenvalue set of the corresponding $$L^2[0,1]$$ eigenvalue problem for such KdV potentials, thus extends Takemura’s works (Commun Math Phys 235:467–494, 2003) and (Electron J Differ Equ 2004(15):1–30, 2004). Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy et al. (Trans Am Math Soc 358:603–656, 2006) and the Darboux transformation.