38 research outputs found
Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials
We consider a class of fully-nonlinear Fermi-Pasta-Ulam (FPU) lattices,
consisting of a chain of particles coupled by fractional power nonlinearities
of order . This class of systems incorporates a classical Hertzian
model describing acoustic wave propagation in chains of touching beads in the
absence of precompression. We analyze the propagation of localized waves when
is close to unity. Solutions varying slowly in space and time are
searched with an appropriate scaling, and two asymptotic models of the chain of
particles are derived consistently. The first one is a logarithmic KdV
equation, and possesses linearly orbitally stable Gaussian solitary wave
solutions. The second model consists of a generalized KdV equation with
H\"older-continuous fractional power nonlinearity and admits compacton
solutions, i.e. solitary waves with compact support. When , we numerically establish the asymptotically Gaussian shape of exact FPU
solitary waves with near-sonic speed, and analytically check the pointwise
convergence of compactons towards the limiting Gaussian profile
Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation
We study nonlinear waves in Newton's cradle, a classical mechanical system
consisting of a chain of beads attached to linear pendula and interacting
nonlinearly via Hertz's contact forces. We formally derive a spatially discrete
modulation equation, for small amplitude nonlinear waves consisting of slow
modulations of time-periodic linear oscillations. The fully-nonlinear and
unilateral interactions between beads yield a nonstandard modulation equation
that we call the discrete p-Schroedinger (DpS) equation. It consists of a
spatial discretization of a generalized Schroedinger equation with p-Laplacian,
with fractional p>2 depending on the exponent of Hertz's contact force. We show
that the DpS equation admits explicit periodic travelling wave solutions, and
numerically find a plethora of standing wave solutions given by the orbits of a
discrete map, in particular spatially localized breather solutions. Using a
modified Lyapunov-Schmidt technique, we prove the existence of exact periodic
travelling waves in the chain of beads, close to the small amplitude modulated
waves given by the DpS equation. Using numerical simulations, we show that the
DpS equation captures several other important features of the dynamics in the
weakly nonlinear regime, namely modulational instabilities, the existence of
static and travelling breathers, and repulsive or attractive interactions of
these localized structures
Periodic Travelling Waves in Dimer Granular Chains
We study bifurcations of periodic travelling waves in granular dimer chains
from the anti-continuum limit, when the mass ratio between the light and heavy
beads is zero. We show that every limiting periodic wave is uniquely continued
with respect to the mass ratio parameter and the periodic waves with the
wavelength larger than a certain critical value are spectrally stable.
Numerical computations are developed to study how this solution family is
continued to the limit of equal mass ratio between the beads, where periodic
travelling waves of granular monomer chains exist
Justification of the log-KdV equation in granular chains: the case of precompression
For travelling waves with nonzero boundary conditions, we justify the
logarithmic Korteweg-de Vries equation as the leading approximation of the
Fermi-Pasta-Ulam lattice with Hertzian nonlinear potential in the limit of
small anharmonicity. We prove control of the approximation error for the
travelling solutions satisfying differential advance-delay equations, as well
as control of the approximation error for time-dependent solutions to the
lattice equations on long but finite time intervals. We also show nonlinear
stability of the travelling waves on long but finite time intervals.Comment: 29 page