Symmetries of the Continuous and Discrete Krichever-Novikov Equation

where P (u) is an arbitrary fourth degree polynomial of its argument with constant coefficients. This is a nonlinear partial differential equation with 5 arbitrary constant parameters. Equation (1.1) first appeared in the study of the finite gap solutions of the Kadomtsev-Petviashvili equation [8,7,21]. For a special choice of P (u) (1.1) reduces to the Korteweg-de Vries equation but for a generic polynomial no differential substitution exists reducing equation (1.1) to a KdV-type equation [24]. In [7,5,20], a zero-curvature representation was obtained for (1.1) involving sl(2) matrices. The Hamiltonian structure of (1.1) was analyzed and possible applications were reviewed in [23,17]. Bäcklund transformations have been constructed together with the nonlinear superposition formulae in [1]. The Lax representation was used in [23] to prove that (1.1) has conservation laws. In [3] the authors considered a generalization of (1.1) in which the polynomial P (u) is an arbitrary function of u and studied its symmetry classification. In 1983 Yamilov [30] introduced an integrable discretization of the Krichever-Novikov equation (the YdKN equation):

where the polynomial S n is given by

This is a differential-difference equation with 6 arbitrary constant parameters. By carrying out the continuous limit, we get the Krichever-Novikov equation (1.1) [30] (see Section 2 below).

The YdKN equation has been obtained as a result of a classification of differential-difference equations of the form un = f (u n-1 , u n , u n+1 ) with no explicit n and t dependence [30,31] that allow at least two conservation laws (or one conservation law and one generalized symmetry) of a high enough order. In the general case, when all parameters are different from zero, (1.2), (1.3) is the only example in the complete list of Volterra type equations which cannot be transformed by Miura transformations into the Volterra or Toda lattice equation [31]. Recently it has been observed that most of the known integrable discrete equations on square lattices are closely related to the YdKN equation in the sense that they generate Bäcklund transformations of the YdKN equation [10,27,16]. An L-A pair for the YdKN equation has been constructed in [27].

A generalization of the YdKN equation (GYdKN) introduced by D. Levi and R. Yamilov in [15] has the same form (1.2), (1.3), but with n-dependent coefficients:

Here β n , γ n , δ n are two-periodic, i.e. can be written in the form

Thus the GYdKN equation depends on 9 arbitrary constant parameters. It has been shown in [15] that the GYdKN equation satisfies the lowest integrability conditions in the generalized symmetry classification of Volterra type equations. Both YdKN and GYdKN equations are integrable in the sense that they possess master symmetries [2] and therefore they have infinite hierarchies of generalized symmetries and conservation laws. The GYdKN equation is also closely related to non-autonomous discrete equations on square lattices [29]. It is worth mentioning here that this generalization does not allow a continuous limit to the Krichever-Novikov equation or any of its generalizations. Extensions of the YdKN, which in the continuous limit reduce to the KN equation or its generalizations can be obtained by choosing P n , Q n and R n as arbitrary t-independent functions of u n . An interesting extension of the YdKN equation is given by the equation

where α, . . . , ω are 9 real constants, at least one of them nonzero. We will call (1.6) the EYdKN (extended YdKN). Like the GYdKN the EYdKN equation depends on 9 constant coefficients. By choosing β = β, γ = γ, δ = δ it reduces to the YdKN equation.

In the following we are going to carry out the point symmetry classification for all particular cases of the EYdKN equation. These are differential-difference equations and for them we will use the theory of symmetries of difference equations as presented in [4,11,13,26,14]. Due to its complication we present here just one example of an equation belonging to the GYdKN class which possesses a nontrivial point symmetry algebra.

In Section 2 we first take the continous limit of a generalized YdKN equation and then calculate the Lie point symmetries of the obtained (continuous) generalized Krichever-Novikov equation (1.1) in which f (u) ≡ P (u) is an arbitrary function. Sections 3 and 4 are devoted to a symmetry classification of the EYdKN equation for which P (u n ), Q(u n ) and R(u n ) are restricted to being second order polynomials. This includes the integrable YdKN equation as a subcase. Some conclusions and future outlook are presented in Section 5.

2 Continuous limit of a generalized YdKN equation and its Lie point symmetries

Let us look for the continuous limit of a generalization of the YdKN equation (1.2), (1.3). Here, for the sake of simplicity of notation we take P (u n ), Q(u n ) and R(u n ) as arbitrary functions of their argument