Prolongation structure of the Krichever-Novikov equation

Sergei Igonine, Ruud Martini

Research output: Contribution to journalArticleAcademicpeer-review

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 languageUndefined
Pages (from-to)9801-9810
Number of pages10
JournalJournal of physics A: mathematical and general
Volume35
Issue number46
DOIs
Publication statusPublished - 2002

Keywords

  • IR-102312
  • METIS-209346

Cite this

Igonine, Sergei ; Martini, Ruud. / Prolongation structure of the Krichever-Novikov equation. In: Journal of physics A: mathematical and general. 2002 ; Vol. 35, No. 46. pp. 9801-9810.
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Prolongation structure of the Krichever-Novikov equation. / Igonine, Sergei; Martini, Ruud.

In: Journal of physics A: mathematical and general, Vol. 35, No. 46, 2002, p. 9801-9810.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Prolongation structure of the Krichever-Novikov equation

AU - Igonine, Sergei

AU - Martini, Ruud

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.

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