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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, United States |
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). Google Scholar |
[2] |
R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations,, Academic, (1983). Google Scholar |
[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). Google Scholar |
[4] |
M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sine-Gordon system,, \emph{Physica D}, 14 (1984), 88. Google Scholar |
[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. Google Scholar |
[6] |
Y. S. Kivshar and D. K. Campbell, Peierls-Nabarro potential for highly localized nonlinear modes, \emph{Phys. Rev. E }, 48 (1993), 3077. Google Scholar |
[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. Google Scholar |
[8] |
C.-M. Marle, Rep. Math. Phys., Various approaches to conservative and nonconservative nonholonomic systems, 42 (1998), 211. Google Scholar |
[9] |
H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys. 47 (2006), 47 (2006). Google Scholar |
[10] |
V. I. Arnold, V. V. Kozlov and A. I. Neistadt, Mathematical Methods of Classical and Celestial Mechanics,, 3rd Ed, (2006). Google Scholar |
[11] |
M. Remoissenet, Waves called solitons,, Springer-Verlag, (1999). Google Scholar |
[12] |
T. Dauxois and M. Peyrard, Physics of Solitons,, Cambridge University Press, (2006). Google Scholar |
[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. Google Scholar |
[14] |
S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201. Google Scholar |
[15] |
P.G. Kevrekidis, Non-linear waves in lattices: past, present, future,, \emph{IMA J. Appl. Math.}, 76 (2011), 389. Google Scholar |
[16] |
D.E. Pelinovsky, Translationally invariant nonlinear Schrödinger lattices, \emph{Nonlinearity}, 19 (2006), 2695. Google Scholar |
[17] |
P.G. Kevrekidis, On a class of discretizations of Hamiltonian nonlinear partial differential equations,, \emph{Physica D}, 183 (2003), 68. Google Scholar |
[18] |
A. A. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics, (2009). Google Scholar |
[19] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics,, Encyclopedia of Mathematical Sciences, (2007). Google Scholar |
[20] |
S. Flach and A.V. Gorbach, Discrete breathersdvances in theory and applications,, Phys. Rep., 467 (2008), 1. Google Scholar |
[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. Google Scholar |
[22] |
J.G. Caputo and M.P. Soerensen, Radial sine-Gordon kinks as sources of fast breathers,, Phys. Rev. E, 88 (2013). Google Scholar |
show all references
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). Google Scholar |
[2] |
R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, and H.C. Morris, Solitons and Nonlinear Wave Equations,, Academic, (1983). Google Scholar |
[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). Google Scholar |
[4] |
M. Peyrard and M.D. Kruskal, Kink dynamics in the highly discrete sine-Gordon system,, \emph{Physica D}, 14 (1984), 88. Google Scholar |
[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. Google Scholar |
[6] |
Y. S. Kivshar and D. K. Campbell, Peierls-Nabarro potential for highly localized nonlinear modes, \emph{Phys. Rev. E }, 48 (1993), 3077. Google Scholar |
[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. Google Scholar |
[8] |
C.-M. Marle, Rep. Math. Phys., Various approaches to conservative and nonconservative nonholonomic systems, 42 (1998), 211. Google Scholar |
[9] |
H. Cendra and S. Grillo, Lagrangian systems with higher order constraints,, J. Math. Phys. 47 (2006), 47 (2006). Google Scholar |
[10] |
V. I. Arnold, V. V. Kozlov and A. I. Neistadt, Mathematical Methods of Classical and Celestial Mechanics,, 3rd Ed, (2006). Google Scholar |
[11] |
M. Remoissenet, Waves called solitons,, Springer-Verlag, (1999). Google Scholar |
[12] |
T. Dauxois and M. Peyrard, Physics of Solitons,, Cambridge University Press, (2006). Google Scholar |
[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. Google Scholar |
[14] |
S. Aubry, Breathers in nonlinear lattices: existence, linear stability and quantization, Physica D, 103 (1997), 201. Google Scholar |
[15] |
P.G. Kevrekidis, Non-linear waves in lattices: past, present, future,, \emph{IMA J. Appl. Math.}, 76 (2011), 389. Google Scholar |
[16] |
D.E. Pelinovsky, Translationally invariant nonlinear Schrödinger lattices, \emph{Nonlinearity}, 19 (2006), 2695. Google Scholar |
[17] |
P.G. Kevrekidis, On a class of discretizations of Hamiltonian nonlinear partial differential equations,, \emph{Physica D}, 183 (2003), 68. Google Scholar |
[18] |
A. A. Bloch, Nonholonomic Mechanics and Control,, Interdisciplinary Applied Mathematics, (2009). Google Scholar |
[19] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics,, Encyclopedia of Mathematical Sciences, (2007). Google Scholar |
[20] |
S. Flach and A.V. Gorbach, Discrete breathersdvances in theory and applications,, Phys. Rep., 467 (2008), 1. Google Scholar |
[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. Google Scholar |
[22] |
J.G. Caputo and M.P. Soerensen, Radial sine-Gordon kinks as sources of fast breathers,, Phys. Rev. E, 88 (2013). Google Scholar |
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