41 research outputs found
A convergent string method: Existence and approximation for the Hamiltonian boundary-value problem
This article studies the existence of long-time solutions to the Hamiltonian
boundary value problem, and their consistent numerical approximation. Such a
boundary value problem is, for example, common in Molecular Dynamics, where one
aims at finding a dynamic trajectory that joins a given initial state with a
final one, with the evolution being governed by classical (Hamiltonian)
dynamics. The setting considered here is sufficiently general so that long time
transition trajectories connecting two configurations can be included, provided
the total energy is chosen suitably. In particular, the formulation
presented here can be used to detect transition paths between two stable basins
and thus to prove the existence of long-time trajectories. The starting point
is the formulation of the equation of motion of classical mechanics in the
framework of Jacobi's principle; a curve shortening procedure inspired by
Birkhoff's method is then developed to find geodesic solutions. This approach
can be viewed as a string method
On the -limit for a non-uniformly bounded sequence of two phase metric functionals
In this study we consider the -limit of a highly oscillatory
Riemannian metric length functional as its period tends to 0. The metric
coefficient takes values in either or where and . We
find that for a large class of metrics, in particular those metrics whose
surface of discontinuity forms a differentiable manifold, the -limit
exists, as in the uniformly bounded case. However, when one attempts to
determine the -limit for the corresponding boundary value problem, the
existence of the -limit depends on the value of . Specifically, we
show that the power is critical in that the -limit exists for , whereas it ceases to exist for . The results here have
applications in both nonlinear optics and the effective description of a
Hamiltonian particle in a discontinuous potential.Comment: 31 pages, 1 figure. Submitte
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.Comment: revised version with extended introduction, improved perturbation
method, and novel uniqueness result; 20 pages, 5 figure
Travelling heteroclinic waves in a Frenkel-Kontorova chain with anharmonic on-site potential
The Frenkel-Kontorova model for dislocation dynamics from 1938 is given by a
chain of atoms, where neighbouring atoms interact through a linear spring and
are exposed to a smooth periodic on-site potential. A dislocation moving with
constant speed corresponds to a heteroclinic travelling wave, making a
transition from one well of the on-site potential to another. The ensuing
system is nonlocal, nonlinear and nonconvex. We present an existence result for
a class of smooth nonconvex on-site potentials. Previous results in mathematics
and mechanics have been limited to on-site potentials with harmonic wells. To
overcome this restriction, we first develop a global centre manifold theory for
anharmonic wave trains, then parametrise the centre manifold to obtain
asymptotically correct approximations to the solution sought, and finally
obtain the heteroclinic wave via a fixed point argument