Necessary integrability conditions for evolutionary lattice equations

Abstract

The structure of solutions is studied for the Lax equation $D_t(G)=[F,G]$ for formal power series with respect to the shift operator. It is proved that if the equation with a given series $F$ of degree $m$ admits a solution $G$ of degree $k$ then it admits, as well, a solution $H$ of degree $m$ such that $H^k=G^m$. This property is used for derivation of necessary integrability conditions for scalar evolutionary lattices.

23 pp

References in corpus (4)

• Generalized symmetry classification of discrete equations of a class depending on twelve parameters
• Integrable evolution equations with constant separant
• Differential-difference equations associated with the fractional Lax operators
• Symmetries of the Continuous and Discrete Krichever-Novikov Equation

Cited by in corpus (8)

• Classification of five-point differential-difference equations
• Classification of five-point differential-difference equations II
• Integrable Möbius invariant evolutionary lattices of second order
• Perturbative Symmetry Approach for Differential-Difference Equations
• Non-invertible transformations for the classification of differential-difference equations
• Integrable 7-point discrete equations and evolution lattice equations of order 2
• Integrability test for evolutionary lattice equations of higher order
• Integrable Modifications of the Ito-Narita-Bogoyavlensky Equation