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Nonholonomic constraints and their impact on discretizations of KleinGordon lattice dynamical models
1.  Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 010034515 
2.  Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton AB T6G 2G1 
3.  Department of Mathematics, University of Illinois at UrbanaChampaign, Urbana, Illinois 618012975, United States 
References:
[1] 
J. CuevasMaraver, P.G. Kevrekidis and F. Williams (Eds.), The sineGordon model and its applications: From pendula and Josephson Junctions to Gravity and HighEnergy Physics,, SpringerVerlag, (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 OneDimensional Diatomic Granular Crystals, \emph{Phys. Rev. Lett.}, 104 (2010). 
[4] 
M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sineGordon system,, \emph{Physica D}, 14 (1984), 88. 
[5] 
S.V. Dmitriev, P.G. Kevrekidis and N. Yoshikawa, Standard nearestneighbour discretizations of KleinGordon models cannot preserve both energy and linear momentum,, J. Phys. A, 39 (2006), 7217. 
[6] 
Y. S. Kivshar and D. K. Campbell, PeierlsNabarro potential for highly localized nonlinear modes, \emph{Phys. Rev. E }, 48 (1993), 3077. 
[7] 
S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, Discrete KleinGordon models with static kinks free of the PeierlsNabarro 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,, SpringerVerlag, (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 timereversible 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, Nonlinear 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 SineGordon equation in two space dimensions,, Physica D, 238 (2009), 1229. 
[22] 
J.G. Caputo and M.P. Soerensen, Radial sineGordon kinks as sources of fast breathers,, Phys. Rev. E, 88 (2013). 
show all references
References:
[1] 
J. CuevasMaraver, P.G. Kevrekidis and F. Williams (Eds.), The sineGordon model and its applications: From pendula and Josephson Junctions to Gravity and HighEnergy Physics,, SpringerVerlag, (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 OneDimensional Diatomic Granular Crystals, \emph{Phys. Rev. Lett.}, 104 (2010). 
[4] 
M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sineGordon system,, \emph{Physica D}, 14 (1984), 88. 
[5] 
S.V. Dmitriev, P.G. Kevrekidis and N. Yoshikawa, Standard nearestneighbour discretizations of KleinGordon models cannot preserve both energy and linear momentum,, J. Phys. A, 39 (2006), 7217. 
[6] 
Y. S. Kivshar and D. K. Campbell, PeierlsNabarro potential for highly localized nonlinear modes, \emph{Phys. Rev. E }, 48 (1993), 3077. 
[7] 
S.V. Dmitriev, P.G. Kevrekidis, N. Yoshikawa, Discrete KleinGordon models with static kinks free of the PeierlsNabarro 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,, SpringerVerlag, (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 timereversible 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, Nonlinear 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 SineGordon equation in two space dimensions,, Physica D, 238 (2009), 1229. 
[22] 
J.G. Caputo and M.P. Soerensen, Radial sineGordon kinks as sources of fast breathers,, Phys. Rev. E, 88 (2013). 
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