Integrable Möbius-invariant evolutionary lattices of second order

Abstract

We solve the classification problem for integrable lattices of the form u ,t = f (u −2 , . . . , u 2 ) under the additional assumption of invariance with respect to the group of linear-fractional transformations. The obtained list contains 5 equations, including 3 new. Difference Miura type substitutions are found which relate these equations with known polynomial lattices. We also present some classification results for the generic lattices.

Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation

“…Here ∂ 2 log g ∂v k ∂v 1 = 0 due to (11), ∂ log g (1) ∂v 1 = 0 due to the definition of g (1) in (11), and ∂ 2 log g (2) ∂v k−1 ∂v 1 = 0 due to (13). Therefore ∂ 2 g ∂v k ∂v = 0, i.e.,…”

“…The decomposition shown in diagram (22) allows one to construct the generalized symmetry for (E29) by using a known symmetry for the INB equation (2). It is even easier to use a symmetry, presented in [26], for its well-known modification (E7).…”

“…Lately, equations of this class have been intensively studied by the generalized symmetry method, see, e.g., [1,2,7,8]. However, the problem of description of all integrable equations of this class is far from complete.…”

“…relating (1) and (2). They transform any solution v n of (1) into a solution u n of (2). Such transformation is explicit in one direction and can be called the discrete substitution by analogy with differential substitutions in the continuous case, see, e.g., [16,17], where such transformations were studied.…”

Algebraic Entropy of a Class of Five-Point Differential-Difference Equations

“…70S05; 39A99. 1 We say that a function is well-defined on the phase space if it is analytic and single-valued.…”