Transport and generation of macroscopically modulated waves in diatomic chains

Previous Article
Comparing the efficiency of numerical techniques for the integration of variational equations
PROC Home
This Issue
Next Article
$L^1$ maximal regularity for the laplacian and applications

2011, 2011(Special): 485-494. doi: 10.3934/proc.2011.2011.485

Transport and generation of macroscopically modulated waves in diatomic chains

1.	Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, D-10117 Berlin

Received August 2010 Revised April 2011 Published October 2011

Abstract
Related Papers

We derive and justify analytically the dynamics of a small macroscopically modulated amplitude of a single plane wave in a nonlinear diatomic chain with stabilizing on-site potentials including the case where a wave generates another wave via self-interaction. More precisely, we show that in typical chains acoustical waves can generate optical but not acoustical waves, while optical waves are always closed with respect to self-interaction.

Keywords: macroscopic modulations, resonance conditions, Diatomic chains.

Mathematics Subject Classification: Primary: 37K60, 34E13; Secondary: 70F45, 35L45.

Citation: Johannes Giannoulis. Transport and generation of macroscopically modulated waves in diatomic chains. Conference Publications, 2011, 2011 (Special) : 485-494. doi: 10.3934/proc.2011.2011.485

[1]	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
[2]	Francisco Odair de Paiva, Humberto Ramos Quoirin. Resonance and nonresonance for p-Laplacian problems with weighted eigenvalues conditions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1219-1227. doi: 10.3934/dcds.2009.25.1219
[3]	Wen-Xin Qin. Modulation of uniform motion in diatomic Frenkel-Kontorova model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3773-3788. doi: 10.3934/dcds.2014.34.3773
[4]	F. J. Lin. Hamiltonian dynamics of atom-diatomic molecule complexes and collisions. Conference Publications, 2007, 2007 (Special) : 655-666. doi: 10.3934/proc.2007.2007.655
[5]	Sergey V. Bolotin. Shadowing chains of collision orbits. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 235-260. doi: 10.3934/dcds.2006.14.235
[6]	Luisa Berchialla, Luigi Galgani, Antonio Giorgilli. Localization of energy in FPU chains. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 855-866. doi: 10.3934/dcds.2004.11.855
[7]	Lukas Neumann, Christian Schmeiser. A kinetic reaction model: Decay to equilibrium and macroscopic limit. Kinetic & Related Models, 2016, 9 (3) : 571-585. doi: 10.3934/krm.2016007
[8]	Pierre Degond, Cécile Appert-Rolland, Julien Pettré, Guy Theraulaz. Vision-based macroscopic pedestrian models. Kinetic & Related Models, 2013, 6 (4) : 809-839. doi: 10.3934/krm.2013.6.809
[9]	Yan Yong, Weiyuan Zou. Macroscopic regularity for the relativistic Boltzmann equation with initial singularities. Kinetic & Related Models, 2019, 12 (5) : 945-967. doi: 10.3934/krm.2019036
[10]	Ciro D'Apice, Rosanna Manzo. A fluid dynamic model for supply chains. Networks & Heterogeneous Media, 2006, 1 (3) : 379-398. doi: 10.3934/nhm.2006.1.379
[11]	Jijiang Sun, Chun-Lei Tang. Resonance problems for Kirchhoff type equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 2139-2154. doi: 10.3934/dcds.2013.33.2139
[12]	Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu. Nonlinear Dirichlet problems with double resonance. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1147-1168. doi: 10.3934/cpaa.2017056
[13]	Leszek Gasiński, Nikolaos S. Papageorgiou. Dirichlet $(p,q)$-equations at resonance. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2037-2060. doi: 10.3934/dcds.2014.34.2037
[14]	D. Bonheure, C. Fabry. A variational approach to resonance for asymmetric oscillators. Communications on Pure & Applied Analysis, 2007, 6 (1) : 163-181. doi: 10.3934/cpaa.2007.6.163
[15]	Philip Korman. Curves of equiharmonic solutions, and problems at resonance. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2847-2860. doi: 10.3934/dcds.2014.34.2847
[16]	Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253
[17]	N. Bellomo, A. Bellouquid. From a class of kinetic models to the macroscopic equations for multicellular systems in biology. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 59-80. doi: 10.3934/dcdsb.2004.4.59
[18]	Michael Herty, Reinhard Illner. Analytical and numerical investigations of refined macroscopic traffic flow models. Kinetic & Related Models, 2010, 3 (2) : 311-333. doi: 10.3934/krm.2010.3.311
[19]	Alexander Vibe, Nicole Marheineke. Modeling of macroscopic stresses in a dilute suspension of small weakly inertial particles. Kinetic & Related Models, 2018, 11 (6) : 1443-1474. doi: 10.3934/krm.2018057
[20]	Tomasz Komorowski, Stefano Olla, Marielle Simon. Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities. Kinetic & Related Models, 2018, 11 (3) : 615-645. doi: 10.3934/krm.2018026

Impact Factor:

American Institute of Mathematical Sciences