### Abstract

Original language | Undefined |
---|---|

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 |
---|---|

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

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

}

*Prolongation structure of the Krichever-Novikov equation*. Memorandum Faculty Mathematical Sciences, no. 1638, Meetkundig en toegepast-algebraisch onderzoek, Enschede.

**Prolongation structure of the Krichever-Novikov equation.** / Igonine, Sergei; Martini, Ruud.

Research output: Book/Report › Report › Professional

TY - BOOK

T1 - Prolongation structure of the Krichever-Novikov equation

AU - Igonine, Sergei

AU - Martini, Ruud

N1 - Imported from MEMORANDA

PY - 2002

Y1 - 2002

N2 - 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.

AB - 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.

KW - MSC-37K30

KW - MSC-35Q53

KW - METIS-208635

KW - MSC-37K10

KW - EWI-3458

KW - IR-65825

M3 - Report

SN - 0169-2690

T3 - Memorandum Faculty Mathematical Sciences

BT - Prolongation structure of the Krichever-Novikov equation

PB - Meetkundig en toegepast-algebraisch onderzoek

CY - Enschede

ER -