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.

在本文中，我们证明，随着N趋于无穷大，所有Korteweg-de Vries方程的代数几何有限间隙解都可以实现为N个孤子解的极限（有关此陈述的确切含义，请参见注释1）。这是由Dyachenko等人发起的原始解决方案框架完成的。（2016）和Zakharov等人。（2016）[25，26]。这个结果的一个暗示是，随着N扩散到无穷大，N-孤子解可以很好地逼近Korteweg-de Vries方程的任何有界周期解。我们还从数值上研究了与代数几何有限间隙解具有相同光谱性质但不是代数几何解的原始解。