American Institute of Mathematical Sciences

2015, 2015(special): 696-704. doi: 10.3934/proc.2015.0696

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, United States

Received  September 2014 Revised  March 2015 Published  November 2015

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.
Citation: Panayotis G. Kevrekidis, Vakhtang Putkaradze, Zoi Rapti. Non-holonomic constraints and their impact on discretizations of Klein-Gordon lattice dynamical models. Conference Publications, 2015, 2015 (special) : 696-704. doi: 10.3934/proc.2015.0696
References:
 [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, (2014). [2] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations,, Academic, (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, \emph{Phys. Rev. Lett.}, 104 (2010). [4] M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sine-Gordon system,, \emph{Physica D}, 14 (1984), 88. [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. [6] Y. S. Kivshar and D. K. Campbell, Peierls-Nabarro potential for highly localized nonlinear modes, \emph{Phys. Rev. E }, 48 (1993), 3077. [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. [8] C.-M. Marle, Rep. Math. Phys., Various approaches to conservative and nonconservative nonholonomic systems, 42 (1998), 211. [9] H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys. 47 (2006), 47 (2006). [10] V. I. Arnold, V. V. Kozlov and A. I. Neistadt, Mathematical Methods of Classical and Celestial Mechanics,, 3rd Ed, (2006). [11] M. Remoissenet, Waves called solitons,, Springer-Verlag, (1999). [12] T. Dauxois and M. Peyrard, Physics of Solitons,, Cambridge University Press, (2006). [13] R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,, \emph{Nonlinearity}, 7 (1994), 1623. [14] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201. [15] P.G. Kevrekidis, Non-linear waves in lattices: past, present, future,, \emph{IMA J. Appl. Math.}, 76 (2011), 389. [16] D.E. Pelinovsky, Translationally invariant nonlinear Schrödinger lattices, \emph{Nonlinearity}, 19 (2006), 2695. [17] P.G. Kevrekidis, On a class of discretizations of Hamiltonian nonlinear partial differential equations,, \emph{Physica D}, 183 (2003), 68. [18] A. A. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics, (2009). [19] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics,, Encyclopedia of Mathematical Sciences, (2007). [20] S. Flach and A.V. Gorbach, Discrete breathersdvances in theory and applications,, Phys. Rep., 467 (2008), 1. [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. [22] J.G. Caputo and M.P. Soerensen, Radial sine-Gordon kinks as sources of fast breathers,, Phys. Rev. E, 88 (2013).

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References:
 [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, (2014). [2] R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations,, Academic, (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, \emph{Phys. Rev. Lett.}, 104 (2010). [4] M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sine-Gordon system,, \emph{Physica D}, 14 (1984), 88. [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. [6] Y. S. Kivshar and D. K. Campbell, Peierls-Nabarro potential for highly localized nonlinear modes, \emph{Phys. Rev. E }, 48 (1993), 3077. [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. [8] C.-M. Marle, Rep. Math. Phys., Various approaches to conservative and nonconservative nonholonomic systems, 42 (1998), 211. [9] H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys. 47 (2006), 47 (2006). [10] V. I. Arnold, V. V. Kozlov and A. I. Neistadt, Mathematical Methods of Classical and Celestial Mechanics,, 3rd Ed, (2006). [11] M. Remoissenet, Waves called solitons,, Springer-Verlag, (1999). [12] T. Dauxois and M. Peyrard, Physics of Solitons,, Cambridge University Press, (2006). [13] R.S. MacKay and S. Aubry, Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators,, \emph{Nonlinearity}, 7 (1994), 1623. [14] S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201. [15] P.G. Kevrekidis, Non-linear waves in lattices: past, present, future,, \emph{IMA J. Appl. Math.}, 76 (2011), 389. [16] D.E. Pelinovsky, Translationally invariant nonlinear Schrödinger lattices, \emph{Nonlinearity}, 19 (2006), 2695. [17] P.G. Kevrekidis, On a class of discretizations of Hamiltonian nonlinear partial differential equations,, \emph{Physica D}, 183 (2003), 68. [18] A. A. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics, (2009). [19] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics,, Encyclopedia of Mathematical Sciences, (2007). [20] S. Flach and A.V. Gorbach, Discrete breathersdvances in theory and applications,, Phys. Rep., 467 (2008), 1. [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. [22] J.G. Caputo and M.P. Soerensen, Radial sine-Gordon kinks as sources of fast breathers,, Phys. Rev. E, 88 (2013).
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