Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation

@article{James2010NonlinearWI,
  title={Nonlinear waves in Newton's cradle and the discrete p-Schroedinger equation},
  author={Guillaume James},
  journal={arXiv: Pattern Formation and Solitons},
  year={2010}
}
  • G. James
  • Published 6 August 2010
  • Physics, Mathematics
  • arXiv: Pattern Formation and Solitons
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… Expand
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References

SHOWING 1-10 OF 85 REFERENCES
Localized waves in nonlinear oscillator chains.
TLDR
This paper employs a center manifold reduction method introduced by Iooss and Kirchgassner in the case of traveling waves, which reduces the problem locally to a finite dimensional reversible differential equation and proves the existence of exact traveling breather solutions superposed on an exponentially small periodic tail. Expand
Standing wave instabilities in a chain of nonlinear coupled oscillators
We consider existence and stability properties of nonlinear spatially periodic or quasiperiodic standing waves (SWs) in one-dimensional lattices of coupled anharmonic oscillators. Specifically, weExpand
Travelling waves in the Fermi-Pasta-Ulam lattice
We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbour (the Fermi-Pasta-Ulam model). A constructiveExpand
Solitary waves on Fermi Pasta Ulam lattices: III. Howland-type Floquet theory
Parts II, III and IV of this series are devoted to proving long time stability of solitary waves in one-dimensional nonintegrable lattices with Hamiltonian with a general nearest-neighbour potentialExpand
Micro-macro transition in the atomic chain via Whitham's modulation equation
The subject matter of this paper is the thermodynamic description of the nonlinear atomic chain with temperature. For this reason we consider special approximate solutions of Newton's equations, inExpand
Existence and modulation of traveling waves in particles chains
We consider an infinite particle chain whose dynamics are governed by the following system of differential equations: where qn(t) is the displacement of the nth particle at time t alongExpand
The nonlinear Schrödinger equation as a macroscopic limit for an oscillator chain with cubic nonlinearities
We consider the nonlinear model of an infinite oscillator chain embedded in a background field. We start from an appropriate modulation ansatz of the space–time periodic solutions to the linearizedExpand
Solitary waves on FPU lattices: I. Qualitative properties, renormalization and continuum limit
This paper is the first in a series to address questions of qualitative behaviour, stability and rigorous passage to a continuum limit for solitary waves in one-dimensional non-integrable latticesExpand
DISPERSIVE EVOLUTION OF PULSES IN OSCILLATOR CHAINS WITH GENERAL INTERACTION POTENTIALS
We study the dispersive evolution of modulated pulses in a nonlinear oscillator chain embedded in a background field. The atoms of the chain interact pairwise with an arbitrary but finite numberExpand
Asymptotic solution for solitary waves in a chain of elastic spheres.
  • A. Chatterjee
  • Physics, Medicine
  • Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1999
TLDR
In this work, n is treated as "slightly" greater than 1, and an asymptotic solution is developed in terms of the associated small parameter, which is substantially more accurate than the presently available approximate solution given by Nesterenko. Expand
...
1
2
3
4
5
...