American Institute of Mathematical Sciences

Advanced
February  2003, 3(1): 105-144. doi: 10.3934/dcdsb.2003.3.105

Geometric solitary waves in a 2D mass-spring lattice

Gero Friesecke 1, and  Karsten Matthies 1,

1. 

Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom, United Kingdom

Received  January 2002 Revised  June 2002 Published  November 2002

The existence of longitudinal solitary waves is shown for the Hamiltonian dynamics of a 2D elastic lattice of particles interacting via harmonic springs between nearest and next nearest neighbours. A contrasting nonexistence result for transversal solitary waves is given. The presence of the longitudinal waves is related to the two-dimensional geometry of the lattice which creates a universal overall anharmonicity.
Keywords: geometric nonlinearity., many-particle systems, solitary waves, Lattice dynamics.
Mathematics Subject Classification: 37K60, 74J35, 74J3.
Citation: Gero Friesecke, Karsten Matthies. Geometric solitary waves in a 2D mass-spring lattice. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 105-144. doi: 10.3934/dcdsb.2003.3.105
[1]

Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895
[2]

Jerry Bona, Hongqiu Chen. Solitary waves in nonlinear dispersive systems. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 313-378. doi: 10.3934/dcdsb.2002.2.313
[3]

Yi He, Gongbao Li. Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity. Mathematical Control & Related Fields, 2016, 6 (4) : 551-593. doi: 10.3934/mcrf.2016016
[4]

David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327
[5]

Nicolas Forcadel, Cyril Imbert, Régis Monneau. Homogenization of some particle systems with two-body interactions and of the dislocation dynamics. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 785-826. doi: 10.3934/dcds.2009.23.785
[6]

Doron Levy, Tiago Requeijo. Modeling group dynamics of phototaxis: From particle systems to PDEs. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 103-128. doi: 10.3934/dcdsb.2008.9.103
[7]

José Raúl Quintero, Juan Carlos Muñoz Grajales. Solitary waves for an internal wave model. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5721-5741. doi: 10.3934/dcds.2016051
[8]

Orlando Lopes. A linearized instability result for solitary waves. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 115-119. doi: 10.3934/dcds.2002.8.115
[9]

Luca Dieci, Cinzia Elia. Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2935-2950. doi: 10.3934/dcdsb.2018112
[10]

Emmanuel Hebey. Solitary waves in critical Abelian gauge theories. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1747-1761. doi: 10.3934/dcds.2012.32.1747
[11]

Yiren Chen, Zhengrong Liu. The bifurcations of solitary and kink waves described by the Gardner equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1629-1645. doi: 10.3934/dcdss.2016067
[12]

H. Kalisch. Stability of solitary waves for a nonlinearly dispersive equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 709-717. doi: 10.3934/dcds.2004.10.709
[13]

John Boyd. Strongly nonlinear perturbation theory for solitary waves and bions. Evolution Equations & Control Theory, 2019, 8 (1) : 1-29. doi: 10.3934/eect.2019001
[14]

Kariane Calta, Thomas A. Schmidt. Infinitely many lattice surfaces with special pseudo-Anosov maps. Journal of Modern Dynamics, 2013, 7 (2) : 239-254. doi: 10.3934/jmd.2013.7.239
[15]

Liping Wang. Arbitrarily many solutions for an elliptic Neumann problem with sub- or supercritical nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (3) : 761-778. doi: 10.3934/cpaa.2010.9.761
[16]

Anca-Voichita Matioc. On particle trajectories in linear deep-water waves. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1537-1547. doi: 10.3934/cpaa.2012.11.1537
[17]

André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135
[18]

Mats Ehrnström, Gabriele Villari. Recent progress on particle trajectories in steady water waves. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 539-559. doi: 10.3934/dcdsb.2009.12.539
[19]

Alejandro B. Aceves, Luis A. Cisneros-Ake, Antonmaria A. Minzoni. Asymptotics for supersonic traveling waves in the Morse lattice. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 975-994. doi: 10.3934/dcdss.2011.4.975
[20]

Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (35)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences