# Algebro-Geometric Finite Gap Solutions to the Korteweg--de Vries Equation as Primitive Solutions

## TL;DR

This paper demonstrates that all algebro-geometric finite gap solutions to the Korteweg--de Vries equation can be approximated by N-soliton solutions as N approaches infinity, using the primitive solution framework.

## Contribution

It establishes a new connection between finite gap solutions and N-soliton solutions via primitive solutions, expanding understanding of solution approximation.

## Key findings

- N-soliton solutions can approximate any bounded periodic solution arbitrarily well
- Primitive solutions with similar spectral properties are numerically studied
- All finite gap solutions are limits of N-soliton solutions as N increases

## Abstract

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 the primitive solution framework initiated by [5,28,31]. 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.