Dinh T. Tran

Closed-form expressions for integrals of traveling wave reductions of integrable lattice equations

We give a method to calculate closed-form expressions in terms of multi-sums of products for integrals of ordinary difference equations which are obtained as traveling wave reductions of integrable partial difference equations. Important ingredients are the staircase method, a non-commutative Vieta formula and certain splittings of the Lax matrices. The method is applied to all equations of the Adler–Bobenko–Suris classification, with the exception of Q4.

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

We study the integrability of mappings obtained as reductions of the discrete Korteweg–de Vries (KdV) equation and of two copies of the discrete potential KdV (pKdV) equation. We show that the mappings corresponding to the discrete KdV equation, which can be derived from the latter, are completely integrable in the Liouville–Arnold sense. The mappings associated with two copies of the pKdV equation are also shown to be integrable.

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

Closed form expressions in terms of multi-sums of products have been given in Tran et al (2009 J. Phys A: Math. Theor. 42 225201) and van der Kamp et al (2007 J. Phys. A: Math. Theor. 39 12789–98) of integrals of sine-Gordon, modified Korteweg–de Vries and potential Korteweg–de Vries maps obtained as so-called (p, −1)-travelling 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.

Sufficient number of integrals for the pth-order Lyness equation

In this communication, we present a sufficient number of first integrals for the Lyness equation of arbitrary order. We first use the staircase method (Quispel et al 1991 Physica A 173 243–66) to construct integrals of a derivative equation of the Lyness equation. Closed-form expressions for the integrals are given based on a non-commutative Vieta expansion. The integrals of the Lyness equation then follow directly from these integrals. Previously found integrals for the Lyness equation arise as special cases of our new set of integrals.

A novel nth order difference equation that may be integrable

We derive an nth order difference equation as a dual of a very simple periodic equation, and construct ⌊(n + 1)/2⌋ explicit integrals and integrating factors of this equation in terms of multi-sums of products. We also present a generating function for the degrees of its iterates, exhibiting polynomial growth. In conclusion we demonstrate how the equation in question arises as a reduction of a system of lattice equations related to an integrable equation of Levi and Yamilov. These three facts combine to suggest the integrability of the nth order difference equation.

Integrable and superintegrable systems associated with multi-sums of products.

We construct and study certain Liouville integrable, superintegrable and non-commutative integrable systems, which are associated with multi-sums of products.

Poisson structures for lifts and periodic reductions of integrable lattice equations

We introduce and study suitable Poisson structures for four dimensional maps derived as lifts and specific periodic reductions of integrable lattice equations. These maps are Poisson with respect to these structures and the corresponding integrals are in involution.

Poisson brackets of mappings obtained as (q,−p) reductions of lattice equations

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The (q,−p) reductions are (p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,−2) reductions of the integrable partial difference equations are Liouville integrable in their own right.

Darboux integrability of determinant and equations for principal minors

We consider equations that represent a constancy condition for a 2D Wronskian, mixed Wronskian-Casoratian and 2D Casoratian. These determinantal equations are shown to have the number of independent integrals equal to their order - this implies Darboux integrability. On the other hand, the recurrent formulas for the leading principal minors are equivalent to the 2D Toda equation and its semi-discrete and lattice analogues with particular boundary conditions (cut-off constraints). This connection is used to obtain recurrent formulas and closed-form expressions for integrals of the Toda-type equations from the integrals of the determinantal equations. General solutions of the equations corresponding to vanishing determinants are given explicitly while in the non-vanishing case they are given in terms of solutions of ordinary linear equations.

SIGNATURES OVER FINITE FIELDS OF GROWTH PROPERTIES FOR LATTICE EQUATIONS

We study integrable lattice equations and their perturbations over finite fields. We write these equations in projective coordinates and assign boundary values along axes in the first quadrant. We propose some growth diagnostics over finite fields that can often distinguish between integrable equations and their non-integrable perturbations. We also discuss the limitations of the diagnostic. Finally, we show that conducting parameter searches over finite fields for lattice equations that satisfy a factorization test leads to potential new equations that have vanishing entropy.