Involutivity of integrals of sine-Gordon, modified KdV and potential KdV maps

Abstract

Closed form expressions in terms of multi-sums of products have been given in [13,16] of integrals of sine-Gordon, modified Korteweg-de Vries and potential Korteweg-de Vries maps obtained as socalled (p, −1)-traveling wave reductions of the corresponding partial difference equations. We prove the involutivity of these integrals with respect to recently found symplectic structures for those maps. The proof is based on explicit formulae for the Poisson brackets between multi-sums of products. Involutivity of sine-Gordon integralsIn this section, we distinguish two cases: the odd-dimensional and even-dimensional sine-Gordon maps. In [16] it is shown that for the even-dimensional map, we have enough integrals for integrability. For the odd-dimensional map, we need to reduce the dimension of the map by one. We expand the Poisson bracket between two integrals {I r , I s } as a quadratic polynomial in the parameters α 1 , α 2 , α 3 and prove the involutivity of integrals (8) by showing its coefficients vanish.

Poisson structures for difference equations

“…Therefore, according to Lemma 1, a constant Poisson structure T is preserved by the pKdV map if and only if the quadratic log-canonical Poisson structure defined by the matrix T is preserved by the mKdV map. This is in accordance with the results of [25]. An easy calculation, using (13) or (15), shows that in even dimensions the SG map preserves the non-degenerate log-canonical Poisson structure defined by the Toeplitz matrix with first line (0, 1).…”

Integrability of reductions of the discrete Korteweg–de Vries and potential Korteweg–de Vries equations

“…In §4, we present closedform expressions for first integrals of each reduced KdV map, and we show that they are in involution with respect to the first of the Poisson structures. This furnishes a direct proof of Liouville integrability for these KdV maps, within the framework of the papers [16,19]. Another proof, based on the pair of compatible Poisson brackets, is also sketched.…”

On some classes of discrete polynomials and ordinary difference equations

“…It is worth remarking that these equations can be obtained as special reductions of partial difference equations for an unknown function of two discrete variables. Studies carried out, for example, in [15], [31] shows that proving the Liouville-Arnold integrability for the map under consideration is a quite difficult and complicated task. Another characteristic of discrete equations claiming to have an adjective 'integrable' is a Lax pair representation.…”