Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models

TL;DR

This paper introduces a novel discretization method for Klein-Gordon lattice models using non-holonomic constraints, which better preserves momentum and enhances the mobility of nonlinear waves compared to standard methods.

Contribution

The paper presents a new discretization approach based on non-holonomic constraints that accurately captures the continuum limit and improves nonlinear wave mobility.

Findings

Discretizations preserve a discrete analogue of momentum.

Energy and momentum conservation cannot be simultaneous for generic data.

Enhanced mobility of nonlinear solutions in the new models.

Abstract

We explore a new type of discretizations of lattice dynamical models of the Klein-Gordon type relevant to the existence and long-term mobility of nonlinear waves. The discretization is based on non-holonomic constraints and is shown to retrieve the "proper" continuum limit of the model. Such discretizations are useful in exactly preserving a discrete analogue of the momentum. It is also shown that for generic initial data, the momentum and energy conservation laws cannot be achieved concurrently. Finally, direct numerical simulations illustrate that our models yield considerably higher mobility of strongly nonlinear solutions than the well-known "standard" discretizations, even in the case of highly discrete systems when the coupling between the adjacent nodes is weak. Thus, our approach is better suited for cases where an accurate description of mobility for nonlinear traveling waves is important.