On computation of Darboux polynomials for full Toda lattice

TL;DR

The paper addresses computing Darboux polynomials and Jacobi multipliers for the full Toda lattice using the Darboux framework, without relying on Lagrangian, Hamiltonian, or spectral input. It applies the method of undetermined coefficients to the polynomial vector field $X$ of the full symmetric Toda lattice and its Kostant-Toda extension, solving linear and bi-linear systems derived from $\partial P = c P$ to extract invariants with cofactors $c_m$ chosen to be linear in diagonal elements after a suitable coordinate change. It demonstrates that the full symmetric case yields $n-2$ independent first integrals and an invariant volume form, while the Kostant-Toda case produces two independent rational first integrals and a Jacobi multiplier, confirming integrability in quadratures. The approach is computationally practical (seconds to minutes on a standard laptop for moderate $N$) and offers a path toward structure-preserving discretizations and future work on discrete Darboux invariants via methods such as the Kahan-Hirota-Kimura scheme.

Abstract

One of the oldest methods for computing invariants of ordinary differential equations is tested using the full Toda lattice model. We show that the standard method of undetermined coefficients and modern symbolic algebra tools together with sufficient computing power allow to compute Darboux invariants without any additional information.