Prolongation structure of the Krichever-Novikov equation

Sergei Igonine, Ruud Martini

Research output: Book/ReportReportProfessional

10 Citations (Scopus)
55 Downloads (Pure)

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
Place of PublicationEnschede
PublisherMeetkundig en toegepast-algebraisch onderzoek
Number of pages12
ISBN (Print)0169-2690
Publication statusPublished - 2002

Publication series

NameMemorandum Faculty Mathematical Sciences
PublisherUniversity 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.
Igonine, Sergei ; Martini, Ruud. / Prolongation structure of the Krichever-Novikov equation. Enschede : Meetkundig en toegepast-algebraisch onderzoek, 2002. 12 p. (Memorandum Faculty Mathematical Sciences; 1638).
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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.",
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Igonine, S & Martini, R 2002, 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.

Enschede : Meetkundig en toegepast-algebraisch onderzoek, 2002. 12 p. (Memorandum Faculty Mathematical Sciences; No. 1638).

Research output: Book/ReportReportProfessional

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.

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KW - EWI-3458

KW - IR-65825

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Igonine S, Martini R. Prolongation structure of the Krichever-Novikov equation. Enschede: Meetkundig en toegepast-algebraisch onderzoek, 2002. 12 p. (Memorandum Faculty Mathematical Sciences; 1638).