American Institute of Mathematical Sciences

Advanced
October  2011, 4(5): 1107-1118. doi: 10.3934/dcdss.2011.4.1107

Breathers and kinks in a simulated crystal experiment

Qingxu Dou 1, , Jesús Cuevas 2, , J. C. Eilbeck 3, and  Francis Michael Russell 4,

1. 

School of Engineering and Physical Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, United Kingdom
2. 

Grupo de Física No Lineal. Departamento de Física Aplicada I., Escuela Politécnica Superior. Universidad de Sevilla, C/ Virgen de África, 7, 41011-Sevilla
3. 

Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, United Kingdom
4. 

Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Riccarton, Edinburgh, EH14 4AS, United Kingdom

Received  September 2009 Revised  December 2009 Published  December 2010

We develop a simple 1D model for the scattering of an incoming particle hitting the surface of mica crystal, the transmission of energy through the crystal by a localized mode, and the ejection of atom(s) at the incident or distant face. This is the first attempt to model the experiment described by Russell and Eilbeck in 2007 (EPL, 78, 10004). Although very basic, the model shows many interesting features, for example a complicated energy dependent transition between breather modes and a kink mode, and multiple ejections at both incoming and distant surfaces. In addition, the effect of a heavier surface layer is modelled, which can lead to internal reflections of breathers or kinks at the crystal surface.
Keywords: Nonlinear vibrational modes, coupled oscillators..
Mathematics Subject Classification: Primary: 70Kxx; Secondary: 70Fxx, 34C15, 34C6.
Citation: Qingxu Dou, Jesús Cuevas, J. C. Eilbeck, Francis Michael Russell. Breathers and kinks in a simulated crystal experiment. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1107-1118. doi: 10.3934/dcdss.2011.4.1107
References:
[1]

J. Cuevas, J. F. R. Archilla, B. Sánchez-Rey and F. R. Romero, Interaction of moving discrete breathers with vacancies,, Physica D, 216 (2006).  doi: 10.1016/j.physd.2005.12.022.  Google Scholar

[2]

J. Cuevas, C. Katerji, J. F. R. Archilla, J. C. Eilbeck and F. M. Russell, Influence of moving breathers on vacancies migration,, Phys. Lett., 315 (2003), 364.  doi: 10.1016/S0375-9601(03)01097-1.  Google Scholar

[3]

J. Cuevas, B. Sánchez-Rey, J. C. Eilbeck and F. M. Russell, Interaction of moving discrete breathers with simulated interstitial defects,, in these proceedings, (2009).   Google Scholar

[4]

S. Flach, Conditions on the existence of localized excitations in nonlinear discrete systems,, Phys. Rev. E, 50 (1994).  doi: 10.1103/PhysRevE.50.3134.  Google Scholar

[5]

S. Flach and A. Gorbach, Discrete breathers - Advances in theory and applications,, Phys. Rep., 267 (2008).  doi: 10.1016/j.physrep.2008.05.002.  Google Scholar

[6]

J. L. Marín, J. C. Eilbeck and F. M. Russell, Localized moving breathers in a 2-D hexagonal lattice,, Phys. Letts. A, 248 (1998), 225.   Google Scholar

[7]

F. M. Russell and J. C. Eilbeck, Evidence for moving breathers in a layered crystal insulator at 300K,, Europhysics Letters, 78 (2007).  doi: 10.1209/0295-5075/78/10004.  Google Scholar

[8]

F. M. Russell and J. C. Eilbeck, Persistent mobile lattice excitations in a crystalline insulator,, in these proceedings, (2009).   Google Scholar

[9]

X. Yi, J. A. D. Wattis, H. Susanto and L. J. Cummings, Discrete breathers in a two-dimensional spring-mass lattice,, J. Phys. A: Math. and Theor., 42 (2009).  doi: 10.1088/1751-8113/42/35/355207.  Google Scholar

show all references

References:
[1]

J. Cuevas, J. F. R. Archilla, B. Sánchez-Rey and F. R. Romero, Interaction of moving discrete breathers with vacancies,, Physica D, 216 (2006).  doi: 10.1016/j.physd.2005.12.022.  Google Scholar

[2]

J. Cuevas, C. Katerji, J. F. R. Archilla, J. C. Eilbeck and F. M. Russell, Influence of moving breathers on vacancies migration,, Phys. Lett., 315 (2003), 364.  doi: 10.1016/S0375-9601(03)01097-1.  Google Scholar

[3]

J. Cuevas, B. Sánchez-Rey, J. C. Eilbeck and F. M. Russell, Interaction of moving discrete breathers with simulated interstitial defects,, in these proceedings, (2009).   Google Scholar

[4]

S. Flach, Conditions on the existence of localized excitations in nonlinear discrete systems,, Phys. Rev. E, 50 (1994).  doi: 10.1103/PhysRevE.50.3134.  Google Scholar

[5]

S. Flach and A. Gorbach, Discrete breathers - Advances in theory and applications,, Phys. Rep., 267 (2008).  doi: 10.1016/j.physrep.2008.05.002.  Google Scholar

[6]

J. L. Marín, J. C. Eilbeck and F. M. Russell, Localized moving breathers in a 2-D hexagonal lattice,, Phys. Letts. A, 248 (1998), 225.   Google Scholar

[7]

F. M. Russell and J. C. Eilbeck, Evidence for moving breathers in a layered crystal insulator at 300K,, Europhysics Letters, 78 (2007).  doi: 10.1209/0295-5075/78/10004.  Google Scholar

[8]

F. M. Russell and J. C. Eilbeck, Persistent mobile lattice excitations in a crystalline insulator,, in these proceedings, (2009).   Google Scholar

[9]

X. Yi, J. A. D. Wattis, H. Susanto and L. J. Cummings, Discrete breathers in a two-dimensional spring-mass lattice,, J. Phys. A: Math. and Theor., 42 (2009).  doi: 10.1088/1751-8113/42/35/355207.  Google Scholar

[1]

Klas Modin, Olivier Verdier. Integrability of nonholonomically coupled oscillators. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1121-1130. doi: 10.3934/dcds.2014.34.1121
[2]

Dan Endres, Martin Kummer. Nonlinear normal modes for the isosceles DST. Conference Publications, 1998, 1998 (Special) : 231-241. doi: 10.3934/proc.1998.1998.231
[3]

Michal Fečkan. Blue sky catastrophes in weakly coupled chains of reversible oscillators. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 193-200. doi: 10.3934/dcdsb.2003.3.193
[4]

R. Yamapi, R.S. MacKay. Stability of synchronization in a shift-invariant ring of mutually coupled oscillators. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 973-996. doi: 10.3934/dcdsb.2008.10.973
[5]

Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891
[6]

Stilianos Louca, Fatihcan M. Atay. Spatially structured networks of pulse-coupled phase oscillators on metric spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3703-3745. doi: 10.3934/dcds.2014.34.3703
[7]

Tetsu Mizumachi, Dmitry Pelinovsky. On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 971-987. doi: 10.3934/dcdss.2012.5.971
[8]

Carlos Garca-Azpeitia, Jorge Ize. Bifurcation of periodic solutions from a ring configuration of discrete nonlinear oscillators. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 975-983. doi: 10.3934/dcdss.2013.6.975
[9]

Wenxian Shen. Global attractor and rotation number of a class of nonlinear noisy oscillators. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 597-611. doi: 10.3934/dcds.2007.18.597
[10]

Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503
[11]

Martina Chirilus-Bruckner, Christopher Chong, Oskar Prill, Guido Schneider. Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 879-901. doi: 10.3934/dcdss.2012.5.879
[12]

Xiaoxue Zhao, Zhuchun Li, Xiaoping Xue. Formation, stability and basin of phase-locking for Kuramoto oscillators bidirectionally coupled in a ring. Networks & Heterogeneous Media, 2018, 13 (2) : 323-337. doi: 10.3934/nhm.2018014
[13]

Ryotaro Tsuneki, Shinji Doi, Junko Inoue. Generation of slow phase-locked oscillation and variability of the interspike intervals in globally coupled neuronal oscillators. Mathematical Biosciences & Engineering, 2014, 11 (1) : 125-138. doi: 10.3934/mbe.2014.11.125
[14]

Richard H. Rand, Asok K. Sen. A numerical investigation of the dynamics of a system of two time-delay coupled relaxation oscillators. Communications on Pure & Applied Analysis, 2003, 2 (4) : 567-577. doi: 10.3934/cpaa.2003.2.567
[15]

Carlo Alabiso, Mario Casartelli. Quasi Normal modes in stochastic domains. Conference Publications, 2003, 2003 (Special) : 21-29. doi: 10.3934/proc.2003.2003.21
[16]

Stephen P. Shipman, Darko Volkov. Existence of guided modes on periodic slabs. Conference Publications, 2005, 2005 (Special) : 784-791. doi: 10.3934/proc.2005.2005.784
[17]

Christopher M. Kribs-Zaleta. Alternative transmission modes for Trypanosoma cruzi . Mathematical Biosciences & Engineering, 2010, 7 (3) : 657-673. doi: 10.3934/mbe.2010.7.657
[18]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933
[19]

Shuangjie Peng, Huirong Pi. Spike vector solutions for some coupled nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2205-2227. doi: 10.3934/dcds.2016.36.2205
[20]

Juncheng Wei, Wei Yao. Uniqueness of positive solutions to some coupled nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1003-1011. doi: 10.3934/cpaa.2012.11.1003

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences