### Abstract

We completely describe Wahlquist-Estabrook prolongation structures (coverings) dependent on $u,\,u_x,\,u_{xx},\,u_{xxx}$ for the Krichever-Novikov equation $u_t=u_{xxx}-3u_{xx}^2/(2u_{x})+p(u)/u_{x}+au_{x}$ in the case when the polynomial $p(u)=4u^3-g_2u-g_3$ has distinct roots. We prove that there is a universal prolongation algebra isomorphic to the direct sum of a commutative $2$-dimensional algebra and a certain subalgebra of the tensor product of $\mathfrak{sl}_2(\mathbb{C})$ with the algebra of regular functions on an affine elliptic curve. This is achieved by identifying this prolongation algebra with the one for the anisotropic Landau-Lifshitz equation. Using these results, we find for the Krichever-Novikov equation a new zero-curvature representation, which is polynomial in the spectral parameter in contrast to the known elliptic ones.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | Meetkundig en toegepast-algebraisch onderzoek |

Number of pages | 12 |

ISBN (Print) | 0169-2690 |

Publication status | Published - 2002 |

### Publication series

Name | Memorandum Faculty Mathematical Sciences |
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Publisher | University of Twente, Department of Applied Mathematics |

No. | 1638 |

ISSN (Print) | 0169-2690 |

### Keywords

- MSC-37K30
- MSC-35Q53
- METIS-208635
- MSC-37K10
- EWI-3458
- IR-65825

## Cite this

Igonine, S., & Martini, R. (2002).

*Prolongation structure of the Krichever-Novikov equation*. (Memorandum Faculty Mathematical Sciences; No. 1638). Enschede: Meetkundig en toegepast-algebraisch onderzoek.