On integrability of Hirota-Kimura type discretizations

Abstract

We give an overview of the integrability of the Hirota-Kimura discretization method applied to algebraically completely integrable (a.c.i.) systems with quadratic vector fields. Along with the description of the basic mechanism of integrability (Hirota-Kimura bases), we provide the reader with a fairly complete list of the currently available results for concrete a.c.i. systems.Comment: 47 pages, some minor change

Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

“…Some of the examples have required the use of relatively large computer memory space and computational time. In a forthcoming paper we will apply the method to a large variety of problems inspired by [6] and other literature.…”

“…In [7], we showed that any Hamiltonian systemẎ = J −1 ∇H in arbitrary dimensions, with constant symplectic structure, discretized by Kahan's method has one preserved modified measure and one modified integral which can be seen to coincide with (5) and with (6) respectively. However, the new algorithm presented here is more general as it can be also applied to systems which are not in the formẎ = J −1 ∇H and can be used to find all rational preserved integrals, given sufficient computing power (see Theorem 1).…”

“…, 6, and the following integrals are preserved q i q k , for i = k, of which four are independent. The choice k = 6 yields the integrals presented in [6].…”

“…The algorithm has been implemented in a symbolic algebra system and the codes are adapted to run on clusters of up to 32 cores with up to 768 GBs of memory. In the last two sections, we shall apply the method to some selected examples which have been taken from [14], [16], [13], and [6]. In most cases the map φ is obtained as the application of Kahan's method (4) to a quadratic differential equation (3).…”

Geometric properties of Kahan's method

“…(i) No other method of the family (9) has a modified Hamiltonian when H is cubic, apart from the known cases a = 0, ± Our results are significant and novel for the study of both the integrability of the mappings produced by Kahan's method and for the study of the geometric properties of Runge-Kutta methods: • First, our results explain the integrability of the map obtained when Kahan's method is applied to some of the examples of [10]: their Eq.…”