### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 009, 18 pages      arXiv:1808.06748      https://doi.org/10.3842/SIGMA.2019.009

### On Reducible Degeneration of Hyperelliptic Curves and Soliton Solutions

Atsushi Nakayashiki
Department of Mathematics, Tsuda University, 2-1-1, Tsuda-Machi, Kodaira, Tokyo, Japan

Received August 27, 2018, in final form January 29, 2019; Published online February 08, 2019

Abstract
In this paper we consider a reducible degeneration of a hyperelliptic curve of genus $g$. Using the Sato Grassmannian we show that the limits of hyperelliptic solutions of the KP-hierarchy exist and become soliton solutions of various types. We recover some results of Abenda who studied regular soliton solutions corresponding to a reducible rational curve obtained as a degeneration of a hyperelliptic curve. We study singular soliton solutions as well and clarify how the singularity structure of solutions is reflected in the matrices which determine soliton solutions.

Key words: hyperelliptic curve; soliton solution; KP hierarchy; Sato Grassmannian.

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