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### Open Access Journals

DCDS-S

Using a variational approximation we study discrete solitons of a
nonlinear Schrödinger lattice with a cubic-quintic nonlinearity.
Using an ansatz with six parameters we are able to approximate
bifurcations of asymmetric solutions connecting site-centered and
bond-centered solutions and resulting in the exchange of their
stability. We show that the numerical and variational
approximations are quite close for solitons of small powers.

DCDS-S

It is the purpose of this paper to prove error estimates
for the approximate description of macroscopic wave packets in
infinite periodic chains of coupled oscillators by modulation equations,
like the Korteweg--de Vries (KdV) or the Nonlinear Schrödinger (NLS)
equation. The proofs are based on a discrete
Bloch wave transform of the underlying infinite-dimensional system of coupled ODEs.
After this transform the existing proof for the associated
approximation theorem for the NLS approximation used for
the approximate description of oscillating wave packets in dispersive PDE systems
transfers almost line for line. In contrast, the proof of the approximation theorem
for the KdV approximation of long waves is less obvious. In a special situation
we prove a first approximation result.

DCDS

We consider the leading order quasicontinuum limits of a one-dimensional
granular medium governed by the Hertz contact law under precompression.
The approximate model which is derived in this limit is justified
by establishing asymptotic bounds for the error with the help of energy estimates.
The continuum model predicts the development of shock waves, which are also studied
in the full system with the aid of numerical simulations.
We also show that existing results
concerning the Nonlinear Schrödinger (NLS) and Korteweg de-Vries (KdV) approximation of FPU models apply directly to a precompressed granular medium in the weakly nonlinear regime.

keywords:
error estimates
,
strongly nonlinear
,
granular crystals
,
Quasicontinuum approximation
,
shocks.

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