Exponentially small splitting of homoclinic orbits of parabolic differential equations under periodic forcing
Karsten Matthies
Homoclinic orbits of semilinear parabolic partial differential equations can split under time-periodic forcing as for ordinary differential equations. The stable and unstable manifold may intersect transverse at persisting homoclinic points. The size of the splitting is estimated to be exponentially small of order exp$(-c/\epsilon)$ in the period $\epsilon$ of the forcing with $\epsilon \rightarrow 0$.
keywords: exponential smallness. semilinear parabolic equations splitting Homoclinic orbit
Geometric solitary waves in a 2D mass-spring lattice
Gero Friesecke Karsten Matthies
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
The derivation of the linear Boltzmann equation from a Rayleigh gas particle model
Karsten Matthies George Stone Florian Theil

A linear Boltzmann equation is derived in the Boltzmann-Grad scaling for the deterministic dynamics of many interacting particles with random initial data. We study a Rayleigh gas where a tagged particle is undergoing hard-sphere collisions with background particles, which do not interact among each other. In the Boltzmann-Grad scaling, we derive the validity of a linear Boltzmann equation for arbitrary long times under moderate assumptions on higher moments of the initial distributions of the tagged particle and the possibly non-equilibrium distribution of the background. The convergence of the empiric dynamics to the Boltzmann dynamics is shown using Kolmogorov equations for associated probability measures on collision histories.

keywords: Derivation Boltzmann equation semigroups Rayleigh gas

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