Stability of the Toda equations related to a perturbed $R_i$ type recurrence relation

TL;DR

This paper investigates the stability of Toda lattice equations derived from a perturbed $R_i$ type recurrence relation, using matrix representations and numerical experiments to analyze their behavior.

Contribution

It introduces a modified $R_I$ recurrence relation with perturbed coefficients and derives the associated Toda lattice equations, providing a new perspective on their stability analysis.

Findings

Derived Toda equations from perturbed recurrence coefficients

Recovered a known Lax pair for the system

Numerical experiments indicate stability characteristics

Abstract

In this manuscript, a modified $R_I$ type recurrence relation is considered whose recurrence coefficients are perturbed by addition or multiplication of a constant. The perturbed system of recurrence coefficients is represented by Toda lattice equations, which are derived. These equations are then represented in a matrix form. With the help of this matrix representation, a known Lax pair is recovered. Inferences about the stability of resulting perturbed system of Toda equations are drawn based on numerical experiments.