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

Panayotis G.  Kevrekidis; Vakhtang Putkaradze; Zoi  Rapti

doi:10.3934/proc.2015.0696

Article Contents

2015, Volume 2015: 696-704. Doi: 10.3934/proc.2015.0696 open access

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

1.
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515
2.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1
3.
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801-2975

Received: September 2014

Revised: March 2015

Published: November 2015

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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.

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Mathematics Subject Classification: Primary: 37K60; Secondary: 81Q05.

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