DCDS-S
Variational approximations of bifurcations of asymmetric solitons in cubic-quintic nonlinear Schrödinger lattices
Christopher Chong Dmitry Pelinovsky
Discrete & Continuous Dynamical Systems - S 2011, 4(5): 1019-1031 doi: 10.3934/dcdss.2011.4.1019
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.
keywords: Variational approximations. Discrete nonlinear Schrödinger equations Bifurcations of discrete solitons
DCDS-S
Rigorous description of macroscopic wave packets in infinite periodic chains of coupled oscillators by modulation equations
Martina Chirilus-Bruckner Christopher Chong Oskar Prill Guido Schneider
Discrete & Continuous Dynamical Systems - S 2012, 5(5): 879-901 doi: 10.3934/dcdss.2012.5.879
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.
keywords: approximation. NLS KdV polyatomic FPU
DCDS
Justification of leading order quasicontinuum approximations of strongly nonlinear lattices
Christopher Chong P.G. Kevrekidis Guido Schneider
Discrete & Continuous Dynamical Systems - A 2014, 34(9): 3403-3418 doi: 10.3934/dcds.2014.34.3403
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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