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On algebro-geometric simply-periodic solutions of the KdV hierarchy

Zhijie Chen Tsinghua University Chang-Shou Lin NTU

Classical Analysis and ODEs mathscidoc:1909.43001

Communications in Mathematical Physics, 374, 111-144, 2020
In this paper, we show that as τ → √−1∞, any zero of the Lam´e function converges to either ∞ or a finite point p satisfying Rep = 1 2 and e2πip 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 Rez = 1 2. We also compute explicitly the eigenvalue setofthecorresponding L2[0,1] eigenvalueproblemforsuchKdVpotentials, thus extends Takemura’s works [26, 27]. Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy, Unterkofler and Weikard [11] and the Darboux transformation.
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@inproceedings{zhijie2020on,
  title={On algebro-geometric simply-periodic solutions of the KdV hierarchy},
  author={Zhijie Chen, and Chang-Shou Lin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190902155306849190478},
  booktitle={ Communications in Mathematical Physics},
  volume={374},
  pages={111-144},
  year={2020},
}
Zhijie Chen, and Chang-Shou Lin. On algebro-geometric simply-periodic solutions of the KdV hierarchy. 2020. Vol. 374. In Communications in Mathematical Physics. pp.111-144. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190902155306849190478.
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