Prolongation structure of the Krichever Novikov equation

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 2 xx /(2u x )+p(u)/u x +au x in the case when the polynomial p(u) = 4u 3 − g 2 u − 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. (2000): 37K10, 37K30, 35Q53 Mathematics Subject Classification

Coverings and fundamental algebras for partial differential equations

“…This is proved by a straightforward computation following the scheme of the proof of Theorem 19. Relation (160) was obtained in [6].…”

Symmetry reductions and traveling wave solutions for the Krichever–Novikov equation

“…In , the author studied the Bäcklund transformation for the KN Equation . In , Igonin and Martini described Wahlquist–Estabrook prolongation structures dependent on u , u x , u xx , u xxx for Equation when f ( u ) is a polynomial with distinct roots, f ( u ) = 4 u 3 + bu + c . In , N. Euler and M. Euler derived solution formulae for the KN equation by a systematic multipotentialization of the equation.…”

“…In [2], the author studied the Bäcklund transformation for the KN Equation (1). In [3], Igonin and Martini described Wahlquist-Estabrook prolongation structures dependent on u, u x , u xx , u xxx for Equation (1) when f .u/ is a polynomial with distinct roots, f .u/ D 4u 3 C bu C c. In [4], N. Euler and M. Euler derived solution formulae for the KN equation by a systematic multipotentialization of the equation. In [5,6], conservation laws and integrability of this equation were studied by Svinolupov et al In [7], Nijhoff presented the Lax pair of Equation (1) and discussed its elliptic form.…”

Classical and nonclassical symmetries for the Krichever-Novikov equation

“…This equation with f (u) = 4u 3 + bu + c, where a, b, c ∈ C, appeared in [5] in connection with a study of finite-gap solutions of the Kadomtsev-Petviashvili equation. The Wahlquist-Estabrook prolongation structures dependent on u, u x , u xx , and u xxx for (2) when f (u) is a polynomial with distinct roots, f (u) = 4u 3 + bu + c, were described in [6]. Solution formulas for the KN equation were derived by a systematic multipotentialization of the equation in [7].…”