Structure preserving noise and dissipation in the Toda lattice

TL;DR

This paper introduces a novel approach to adding structure-preserving noise and dissipation to the Toda lattice using Lie algebra and group theory, leading to new insights into stochastic deformations and their continuum limits.

Contribution

It develops a mathematical framework for structure-preserving stochastic deformations of the Toda lattice based on coadjoint orbits and Lie algebra techniques.

Findings

Preserves coadjoint orbit structure under noise and dissipation.

Derives a stochastic Burgers equation as the continuum limit.

Explores properties of the deformations without numerical simulations.

Abstract

In this paper, we use Flaschka's change of variables of the open Toda lattice and its interpretation in term of the group structure of the LU factorisation as a coadjoint motion on a certain dual of Lie algebra to implement a structure preserving noise and dissipation. Both preserve the structure of coadjoint orbit, that is the space of symmetric tri-diagonal matrices and arise as a new type of multiplicative noise and nonlinear dissipation of the Toda lattice. We investigate some of the properties of these deformations and in particular the continuum limit as a stochastic Burger equation with a nonlinear viscosity. This work is meant to be exploratory, and open more questions that we can answer with simple mathematical tools and without numerical simulations.