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Prolongation structure of the Krichever-Novikov equation

Sergei Igonin·Ruud Martini
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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 sl_2(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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Sergei Igonin, Ruud Martini. 2002-10-01. Prolongation structure of the Krichever-Novikov equation. https://arxiv.org/abs/nlin/0208006

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