### 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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Pages (from-to) | 9801-9810 |

Number of pages | 10 |

Journal | Journal of physics A: mathematical and general |

Volume | 35 |

Issue number | 46 |

DOIs | |

Publication status | Published - 2002 |

### Keywords

- IR-102312
- METIS-209346

## Cite this

Igonine, S., & Martini, R. (2002). Prolongation structure of the Krichever-Novikov equation.

*Journal of physics A: mathematical and general*,*35*(46), 9801-9810. https://doi.org/10.1088/0305-4470/35/46/306