Homogenization of accelerated Frenkel-Kontorova models with $n$ types of particles

We consider systems of ODEs that describe the dynamics of particles. Each particle satisfies a Newton law (including a damping term and an acceleration term) where the force is created by the interactions with other particles and with a periodic potential. The presence of a damping term allows the system to be monotone. Our study takes into account the fact that the particles can be different. After a proper hyperbolic rescaling, we show that solutions of these systems of ODEs converge to solutions of some macroscopic homogenized Hamilton-Jacobi equations.

[1]  Gary J. Slater,et al.  Concepts, Methods, and Applications , 2011 .

[2]  S. Lenhart.  Viscosity solutions for weakly coupled systems of first-order partial differential equations , 1988 .

[3]  R. MacKay,et al.  A novel preserved partial order for cooperative networks of units with overdamped second order dynamics, and application to tilted Frenkel–Kontorova chains , 2004 .

[4]  F. Camilli,et al.  HOMOGENIZATION OF MONOTONE SYSTEMS OF HAMILTON-JACOBI EQUATIONS , 2008, 0802.0932.

[5]  L. Evans.  The perturbed test function method for viscosity solutions of nonlinear PDE , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Yuri S. Kivshar,et al.  The Frenkel-Kontorova Model: Concepts, Methods, and Applications , 2004 .

[7]  L. Evans,et al.  Optimal Switching for Ordinary Differential Equations , 1984 .

[8]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[9]  Hitoshi Ishii,et al.  Perron's method for monotone systems of second-order elliptic partial differential equations , 1992, Differential and Integral Equations.

[10]  M. Jazar,et al.  Dynamics of Dislocation Densities in a Bounded Channel. Part II: Existence of Weak Solutions to a Singular Hamilton–Jacobi/Parabolic Strongly Coupled System , 2009, 0903.1435.

[11]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[12]  M. Jazar,et al.  Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system , 2009, 0903.0964.

[13]  Régis Monneau,et al.  Homogenization of some particle systems with two-body interactions and of the dislocation dynamics , 2008 .

[14]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[15]  Régis Monneau,et al.  Homogenization of fully overdamped Frenkel-Kontorova models , 2009 .

[16]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[17]  S. Lenhart.  Viscosity Solutions for Weakly Coupled Systems of First Order PDEs , 1986 .

[18]  LA Rafaelde.  KAM THEORY FOR EQUILIBRIUM STATES IN 1-D STATISTICAL MECHANICS MODELS , 2005 .

[19]  Y. Sire,et al.  Communications in Mathematical Physics Travelling Breathers with Exponentially Small Tails in a Chain of Nonlinear Oscillators , 2022 .

[20]  Yuri S. Kivshar,et al.  The Frenkel-Kontorova Model , 2004 .

[21]  Régis Monneau,et al.  Homogenization of First-Order Equations with $$(u/\varepsilon)$$ -Periodic Hamiltonians. Part I: Local Equations , 2007 .

[22]  H. Ishii,et al.  Viscosity solutions for monotone systems of second-order elliptic PDES , 1991 .

[23]  Régis Monneau,et al.  Homogenization of First Order Equations with (u/ϵ)-Periodic Hamiltonians Part II: Application to Dislocations Dynamics , 2008 .

[24]  S. Aubry.  The twist map, the extended Frenkel-Kontorova model and the devil's staircase , 1983 .

[25]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[26]  R. Llave.  KAM Theory for Equilibrium States in 1-D Statistical Mechanics Models , 2008 .