Generating a chain of maps which preserve the same integral as a given map

Abstract

We generalise the concept of duality to systems of ordinary difference equations (or maps). We propose a procedure to construct a chain of systems of equations which are dual, with respect to an integral H, to the given system, by exploiting the integral relation, defined by the upshifted version and the original version of H. When the numerator of the integral relation is biquadratic or multi-linear, we point out conditions where a dual fails to exists. The procedure is applied to several two-component systems obtained as periodic reductions of 2D lattice equations, including the nonlinear Schrödinger system, the two-component potential Korteweg-De Vries equation, the scalar modified Korteweg-De Vries equation, and a modified Boussinesq system.

Dynamics of A Re-Parametrization of A 2-Dimensional Mapping Derived from Double Discrete Sine-Gordon Mapping

“…This works beautifully for a single discrete equation, although the resulting equation may not be new nor integrable. This idea of dual is extended to system of discrete equations in (Tuwankotta et al, 2019). The latter is interesting in the sense that the method proposed there produces in general more than one system.…”