American Institute of Mathematical Sciences

Advanced
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

Panayotis G. Kevrekidis 1, , Vakhtang Putkaradze 2, and  Zoi Rapti 3,

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.
Keywords: discrete solitons, conservation laws., kinks, Peierls-Nabarro barrier, Klein-Gordon models.
Mathematics Subject Classification: Primary: 37K60; Secondary: 81Q0.
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, 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.

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, 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.

[1]

Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224
[2]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic and Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35
[3]

Christophe Chalons, Paola Goatin, Nicolas Seguin. General constrained conservation laws. Application to pedestrian flow modeling. Networks and Heterogeneous Media, 2013, 8 (2) : 433-463. doi: 10.3934/nhm.2013.8.433
[4]

Andrew Comech. Weak attractor of the Klein-Gordon field in discrete space-time interacting with a nonlinear oscillator. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2711-2755. doi: 10.3934/dcds.2013.33.2711
[5]

Olivier Goubet, Marilena N. Poulou. Semi discrete weakly damped nonlinear Klein-Gordon Schrödinger system. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1525-1539. doi: 10.3934/cpaa.2014.13.1525
[6]

Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010
[7]

Marcelo M. Cavalcanti, Leonel G. Delatorre, Daiane C. Soares, Victor Hugo Gonzalez Martinez, Janaina P. Zanchetta. Uniform stabilization of the Klein-Gordon system. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5131-5156. doi: 10.3934/cpaa.2020230
[8]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903
[9]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076
[10]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679
[11]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359
[12]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[13]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete and Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215
[14]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251
[15]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233
[16]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973
[17]

Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315
[18]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
[19]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448
[20]

Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389

 Impact Factor: 

Tools

Metrics

  • PDF downloads (139)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy