\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract / Introduction Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 37K60; Secondary: 81Q05.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    J. Cuevas-Maraver, P.G. Kevrekidis and F. Williams (Eds.), The sine-Gordon model and its applications: From pendula and Josephson Junctions to Gravity and High-Energy Physics, Springer-Verlag, Heidelberg, 2014.

    [2]

    R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations, Academic, New York, 1983.

    [3]

    N. Boechler, G. Theocharis, S. Job, P. G. Kevrekidis, M. A. Porter, and C. Daraio, Discrete Breathers in One-Dimensional Diatomic Granular Crystals Phys. Rev. Lett., 104, (2010), 244302.

    [4]

    M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sine-Gordon system, Physica D, 14, (1984), 88-102.

    [5]

    S.V. Dmitriev, P.G. Kevrekidis and N. Yoshikawa, Standard nearest-neighbour discretizations of Klein-Gordon models cannot preserve both energy and linear momentum, J. Phys. A, 39, (2006), 7217-7226.

    [6]

    Y. S. Kivshar and D. K. Campbell, Peierls-Nabarro potential for highly localized nonlinear modes Phys. Rev. E , 48, (1993), 3077-3081.

    [7]

    S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, Discrete Klein-Gordon models with static kinks free of the Peierls-Nabarro potential, J. Phys. A, 38, (2005), 7617-7627.

    [8]

    C.-M. Marle, Rep. Math. Phys. Various approaches to conservative and nonconservative nonholonomic systems, 42 (1998) , 211-229.

    [9]

    H. Cendra and S. Grillo, Lagrangian systems with higher order constraints, J. Math. Phys. 47 (2006), 022902.

    [10]

    V. I. Arnold, V. V. Kozlov and A. I. Neistadt, Mathematical Methods of Classical and Celestial Mechanics, 3rd Ed, Springer (2006).

    [11]

    M. Remoissenet, Waves called solitons, Springer-Verlag, Berlin, 1999.

    [12]

    T. Dauxois and M. Peyrard, Physics of Solitons, Cambridge University Press, Cambridge, 2006.

    [13]

    R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators, Nonlinearity, 7, (1994), 1623-1643.

    [14]

    S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization Physica D, 103, (1997), 201-250.

    [15]

    P.G. Kevrekidis, Non-linear waves in lattices: past, present, future, IMA J. Appl. Math., 76, (2011), 389-423.

    [16]

    D.E. Pelinovsky, Translationally invariant nonlinear Schrödinger lattices Nonlinearity, 19, (2006), 2695-2715.

    [17]

    P.G. Kevrekidis, On a class of discretizations of Hamiltonian nonlinear partial differential equations, Physica D, 183, (2003), 68-86.

    [18]

    A. A. Bloch, Nonholonomic Mechanics and Control, Interdisciplinary Applied Mathematics, Springer, 2009.

    [19]

    V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Encyclopedia of Mathematical Sciences, Springer, 2007.

    [20]

    S. Flach and A.V. Gorbach, Discrete breathersdvances in theory and applications, Phys. Rep., 467, (2008), 1-116.

    [21]

    L.A. Cisneros, J. Ize and A.A. Minzoni, Modulational and numerical solutions for the steady discrete Sine-Gordon equation in two space dimensions, Physica D, 238, (2009), 1229-1240.

    [22]

    J.G. Caputo and M.P. Soerensen, Radial sine-Gordon kinks as sources of fast breathers, Phys. Rev. E, 88, (2013), 022915.

  • 加载中
Open Access Under a Creative Commons license
SHARE

Article Metrics

HTML views() PDF downloads(147) Cited by(0)

Access History

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return