Numerical simulations of magnetic billiards in a convex domain in $\mathbb{R}^2$
We present numerical simulations of magnetic billiards inside a convex domain in the plane.
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arXiv subjects
Michael Herrmann
Publications and source records attributed to Michael Herrmann.
Uniqueness of solitary waves in the high-energy limit of FPU-type chains
Recent asymptotic results by the authors provided detailed information on the shape of solitary high-energy travelling waves in FPU atomic chains. In this note we use and extend the methods to understand the linearisation of the travelling wave equation. We show that there are not any other zero eigenvalues than those created by the translation symmetry and this implies a local uniqueness result. The key argument in our asymptotic analysis is to replace the linear advance-delay-differential equation for the eigenfunctions by an approximate ODE.
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High-energy waves in superpolynomial FPU-type chains
We consider periodic traveling waves in FPU-type chains with superpolynomial interaction forces and derive explicit asymptotic formulas for the high-energy limit as well as bounds for the corresponding approximation error. In the proof we adapt twoscale techniques that have recently been develop by Herrmann and Matthies for chains with singular potential and provide an asymptotic ODE for the scaled distance profile.
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Instabilities and oscillations in coagulation equations with kernels of homogeneity one
We discuss the long-time behaviour of solutions to Smoluchowski's coagulation equation with kernels of homogeneity one, combining formal asymptotics, heuristic arguments based on linearization, and numerical simulations. The case of what we call diagonally dominant kernels is particularly interesting. Here one expects that the long-time behaviour is, after a suitable change of variables, the same as for the Burgers equation. However, for kernels that are close to the diagonal one we obtain instability of both, constant solutions and traveling waves and in general no convergence to N-waves for integrable data. On the other hand, for kernels not close to the diagonal one these structures are stable, but the traveling waves have strong oscillations. This has implications on the approach towards an N-wave for integrable data, which is also characterized by strong oscillations near the shock front.
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KdV waves in atomic chains with nonlocal interactions
We consider atomic chains with nonlocal particle interactions and prove the existence of near-sonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.
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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.
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Rate-independent dynamics and Kramers-type phase transitions in nonlocal Fokker-Planck equations with dynamical control
The hysteretic behavior of many-particle systems with non-convex free energy can be modeled by nonlocal Fokker-Planck equations that involve two small parameters and are driven by a time- dependent constraint. In this paper we consider the fast reaction regime related to Kramers-type phase transitions and show that the dynamics in the small-parameter limit can be described by a rate-independent evolution equation with hysteresis. For the proof we first derive mass-dissipation estimates by means of Muckenhoupt constants, formulate conditional stability estimates, and char- acterize the mass flux between the different phases in terms of moment estimates that encode large deviation results. Afterwards we combine all these partial results and establish the dynamical sta- bility of localized peaks as well as sufficiently strong compactness results for the basic macroscopic quantities.
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Interface dynamics in discrete forward-backward diffusion equations
We study the motion of phase interfaces in a diffusive lattice equation with bistable nonlinearity and derive a free boundary problem with hysteresis to describe the macroscopic evolution in the parabolic scaling limit. The first part of the paper deals with general bistable nonlinearities and is restricted to numerical experiments and heuristic arguments. We discuss the formation of macroscopic data and present numerical evidence for pinning, depinning, and annihilation of interfaces. Afterwards we identify a generalized Stefan condition along with a hysteretic flow rule that characterize the dynamics of both standing and moving interfaces. In the second part, we rigorously justify the limit dynamics for single-interface data and a special piecewise affine nonlinearity. We prove persistence of such data, derive upper bounds for the macroscopic interface speed, and show that the macroscopic limit can indeed be described by the free boundary problem. The fundamental ingredient to our proofs is a representation formula that links the solutions of the nonlinear lattice to the discrete heat kernel and enables us to derive macroscopic compactness results in the space of continuous functions.
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Subsonic phase transition waves in bistable lattice models with small spinodal region
Phase transitions waves in atomic chains with double-well potential play a fundamental role in materials science, but 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 localised with respect to the strain variable. As a standard Lyapunov-Schmidt reduction cannot be used due to the presence of an essential spectrum, we characterise the perturbation of the wave as a fixed point of a nonlinear and nonlocal operator and show that this operator is contractive in 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 relation.
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Ground waves in atomic chains with bi-monomial double-well potential
Ground waves in atomic chains are traveling waves that corresponds to minimal non-trivial critical values of the underlying action functional. In this paper we study FPU-type chains with bi-monomial double-well potential and prove the existence of both periodic and solitary ground waves. To this end we minimize the action on the Nehari manifold and show that periodic ground waves converge to solitary ones. Finally, we compute ground waves numerically by a suitable discretization of a constrained gradient flow.
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Kramers and non-Kramers Phase Transitions in Many-Particle Systems with Dynamical Constraint
We study a Fokker-Planck equation with double-well potential that is nonlocally driven by a dynamical constraint and involves two small parameters. Relying on formal asymptotics we identify several parameter regimes and derive reduced dynamical models for different types of phase transitions.
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Self-similar Solutions to a Kinetic Model for Grain Growth
We prove the existence of self-similar solutions to the Fradkov model for two-dimensional grain growth, which consists of an infinite number of nonlocally coupled transport equations for the number densities of grains with given area and number of neighbours (topological class). For the proof we introduce a finite maximal topological class and study an appropriate upwind-discretization of the time dependent problem in self-similar variables. We first show that the resulting finite dimensional differential system has nontrivial steady states. Afterwards we let the discretization parameter tend to zero and prove that the steady states converge to a compactly supported self-similar solution for a Fradkov model with finitely many equations. In a third step we let the maximal topology class tend to infinity and obtain self-similar solutions to the original system that decay exponentially. Finally, we use the upwind discretization to compute self-similar solutions numerically.
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Homoclinic standing waves in focussing DNLS equations --Variational approach via constrained optimization
We study focussing discrete nonlinear Schrödinger equations and present a new variational existence proof for homoclinic standing waves (bright solitons). Our approach relies on the constrained maximization of an energy functional and provides the existence of two one-parameter families of waves with unimodal and even profile function for a wide class of nonlinearities. Finally, we illustrate our results by numerical simulations.
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On selection criteria for problems with moving inhomogeneities
We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Studying simple examples for lattice motion we then compare the implications for both resting and moving inhomogeneities.
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Oscillatory Waves in Discrete Scalar Conservation Laws
We study Hamiltonian difference schemes for scalar conservation laws with monotone flux function and establish the existence of a three-parameter family of periodic travelling waves (wavetrains). The proof is based on an integral equation for the dual wave profile and employs constrained maximization as well as the invariance properties of a gradient flow. We also discuss the approximation of wavetrains and present some numerical results.
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On a Model for Mass Aggregation with Maximal Size
We study a kinetic mean-field equation for a system of particles with different sizes, in which particles are allowed to coagulate only if their sizes sum up to a prescribed time-dependent value. We prove well-posedness of this model, study the existence of self-similar solutions, and analyze the large-time behavior mostly by numerical simulations. Depending on the parameter $\Dconst$, which controls the probability of coagulation, we observe two different scenarios: For $\Dconst>2$ there exist two self-similar solutions to the mean field equation, of which one is unstable. In numerical simulations we observe that for all initial data the rescaled solutions converge to the stable self-similar solution. For $\Dconst<2$, however, no self-similar behavior occurs as the solutions converge in the original variables to a limit that depends strongly on the initial data. We prove rigorously a corresponding statement for $\Dconst\in (0,1/3)$. Simulations for the cross-over case $\Dconst=2$ are not completely conclusive, but indicate that, depending on the initial data, part of the mass evolves in a self-similar fashion whereas another part of the mass remains in the small particles.
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Kramers' formula for chemical reactions in the context of Wasserstein gradient flows
We derive Kramers' formula as singular limit of the Fokker-Planck equation with double-well potential. The convergence proof is based on the Rayleigh principle of the underlying Wasserstein gradient structure and complements a recent result by Peletier, Savaré and Veneroni.
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Heteroclinic standing waves in defocussing DNLS equations -- Variational approach via energy minimization
We study heteroclinic standing waves (dark solitons) in discrete nonlinear Schrödinger equations with defocussing nonlinearity. Our main result is a quite elementary existence proof for waves with monotone and odd profile, and relies on minimizing an appropriately defined energy functional. We also study the continuum limit and the numerical approximation of standing waves.
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