Karsten Matthies

Atomic-scale localization of high-energy solitary waves on lattices

One-dimensional monatomic lattices with Hamiltonian H= P n2 Z Z ( 1 p 2 + V (qn+1 − qn)) are known to carry localized travelling wave solutions, for generic nonlinear potentials V [FW94]. In this paper we derive the asymptotic profile of these waves in the high-energy limit H ! 1 , for LennardJones type interactions. The limit profile is proved to be a universal, highly discrete, piecewise linear wave concentrated on a single atomic spacing. This shows that dispersionless energy transport in these systems is not confined to the long-wave regime on which the theoretical literature has hitherto focused, but also occurs at atomic-scale localization.

Exponential averaging for Hamiltonian evolution equations

We derive estimates on the magnitude of non-adiabatic interaction between a Hamiltonian partial differential equation and a high-frequency nonlinear oscillator. Assuming spatial analyticity of the initial conditions, we show that the dynamics can be transformed to the uncoupled dynamics of an infinite-dimensional Hamiltonian system and an anharmonic oscillator, up to coupling terms which are exponentially small in a certain power of the frequency of the oscillator. The result is derived from an abstract averaging theorem for infinite-dimensional analytic evolution equations in Gevrey spaces. Refining upon a similar result by Neishtadt for analytic ordinary differential equations, the temporal estimate crucially depends on the spatial regularity of the initial condition. The result shows to what extent the strong resonances between rapid forcing and highly oscillatory spatial modes can be suppressed by the choice of sufficiently smooth initial data. An application is provided by a system of nonlinear Schrodinger equations, coupled to a rapidly forcing single mode, representing small-scale oscillations. We provide an example showing that the estimates for partial differential equations we derive here are necessarily different from those in the context of ordinary differential equations.

Subsonic Phase Transition Waves in Bistable Lattice Models with Small Spinodal Region

Although phase transition waves in atomic chains with double-well potential play a fundamental role in materials science, very little is known about their mathematical properties. In particular, the only available results about waves with large amplitudes concern chains with piecewise-quadratic pair potential. In this paper we consider perturbations of a bi-quadratic potential and prove that the corresponding three-parameter family of waves persists as long as the perturbation is small and localized with respect to the strain variable. As a standard Lyapunov--Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterize the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive on a small ball in a suitable function space. Moreover, we derive a uniqueness result for phase transition waves with certain properties and discuss the kinetic relations.

EXPONENTIAL HOMOGENIZATION OF LINEAR SECOND ORDER ELLIPTIC PDEs WITH PERIODIC COEFFICIENTS

A problem of homogenization of a divergence‐type second order uniformly elliptic operator is considered with arbitrary bounded rapidly oscillating periodic coefficients, either with periodic “outer” boundary conditions or in the whole space. It is proved that if the right‐hand side is Gevrey regular (in particular, analytic), then by optimally truncating the full two‐scale asymptotic expansion for the solution one obtains an approximation with an exponentially small error. The optimality of the exponential error bound is established for a one‐dimensional example by proving the analogous lower bound.

Wave pinning in strips

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Can equations of equilibrium predict all physical equilibria? A case study from Field Dislocation Mechanics

Numerical solutions of a one-dimensional model of screw dislocation walls (twist boundaries) are explored. The model is an exact reduction of the three-dimensional system of partial differential equations of Field Dislocation Mechanics. It shares features of both Ginzburg–Landau (GL)-type gradient flow equations and hyperbolic conservation laws, but is qualitatively different from both. We demonstrate such similarities and differences in an effort to understand the equation through simulation. A primary result is the existence of spatially non-periodic, extremely slowly evolving (quasi-equilibrium) cell-wall dislocation microstructures practically indistinguishable from equilibria, which however cannot be solutions to the equilibrium equations of the model, a feature shared with certain types of GL equations. However, we show that the class of quasi-equilibria comprising a spatially non-periodic microstructure consisting of fronts is larger than that of the GL equations associated with the energy of the model. In addition, under applied strain-controlled loading, a single dislocation wall is shown to be capable of moving as a localized entity, as expected in a physical model of dislocation dynamics, in contrast to the associated GL equations. The collective evolution of the quasi-equilibrium cell-wall microstructure exhibits a yielding-type behavior as bulk plasticity ensues, and the effective stress–strain response under loading is found to be rate-dependent. The numerical scheme employed is non-conventional, since wave-type behavior has to be accounted for, and interesting features of two different schemes are discussed. Interestingly, a stable scheme conjectured by us to produce a non-physical result in the present context nevertheless suggests a modified continuum model that appears to incorporate apparent intermittency.

Asymptotic formulas for solitary waves in the high-energy limit of FPU-type chains

It is well established that the solitary waves of FPU-type chains converge in the high-energy limit to traveling waves of the hard-sphere model. In this paper we establish improved asymptotic expressions for the wave profiles as well as an explicit formula for the wave speed. The key step in our approach is the derivation of an asymptotic ODE for the appropriately rescaled strain profile.

A Semigroup Approach to the Justification of Kinetic Theory

This paper develops a method to rigorously show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. We consider a hard sphere flow where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density

Small planar travelling waves in two-dimensional networks of coupled oscillators

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Exponential Estimates in Averaging and Homogenisation

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