Algebro-geometric finite gap solutions to the Korteweg-de Vries equation as primitive solutions

[摘要] In this paper we show that all algebro-geometric finite gap solutions to the Korteweg-de Vries equation can be realized as a limit of N-soliton solutions as N diverges to infinity (see remark 1 for the precise meaning of this statement). This is done using the primitive solution framework initiated by Dyachenko et al. (2016) and Zakharov et al. (2016) [25, 26]. One implication of this result is that the N-soliton solutions can approximate any bounded periodic solution to the Korteweg-de Vries equation arbitrarily well in the limit as N diverges to infinity. We also study primitive solutions numerically that have the same spectral properties as the algebro-geometric finite gap solutions but are not algebro-geometric solutions. (c) 2020 Elsevier B.V. All rights reserved.

[发布日期] 2020-12-15

[关键词] The Korteweg-de Vries equation;Finite gap potentials and solutions;Primitive potentials and solutions;Multi-soliton solutions