## Abstract

We study a family of Korteweg-deVries equations as some evolution equations associated to the Adler-Gelfand-Dikii (AGD) space. First we derive (formally) the Korteweg-deVries (KdV) as an evolution equation of the AGD operator under the action of Vect(S^{1}).The solutions of the AGD operator define an immersion C → ℂP^{n-1} in homogeneous coordinates. We derive the Schwarzian KdV equation as an evolution of the solution curve associated to Δ^{(n)}= d^{n}/dx^{n}+u_{n-2} d^{n-2}/dx^{n-2} + ... + u_{0}. This equation is invariant under linear fractional transformations. We also show how the modified KdV is related to the Schwarzian KdV by the Cole-Hopf transformation. The geometrical (differential Galois theory) connections between all these equations are given.

Original language | British English |
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Pages (from-to) | 17-31 |

Number of pages | 15 |

Journal | Letters in Mathematical Physics |

Volume | 55 |

Issue number | 1 |

State | Published - Jan 2001 |

## Keywords

- AGD space
- Galois theory
- Integrable systems
- KdV equations

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