Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain

Butt, Imran A.; Wattis, Jonathan A.D.

Abstract

We find approximations to travelling breather solutions of the
one-dimensional Fermi-Pasta-Ulam (FPU) lattice. Both bright
breather and dark breather solutions are found. We find that the
existence of localised (bright) solutions depends upon the
coefficients of cubic and quartic terms of the potential energy,
generalising an earlier inequality derived by James [CR Acad Sci
Paris 332, 581, (2001)]. We use the method of multiple scales to
reduce the equations of motion for the lattice to a nonlinear
Schr{\"o}dinger equation at leading order and hence construct an
asymptotic form for the breather. We show that in the absence of
a cubic potential energy term, the lattice supports combined
breathing-kink waveforms. The amplitude of breathing-kinks can be
arbitrarily small, as opposed to traditional monotone kinks, which
have a nonzero minimum amplitude in such systems. We also present
numerical simulations of the lattice, verifying the shape and
velocity of the travelling waveforms, and confirming the
long-lived nature of all such modes.

Citation

Butt, I. A., & Wattis, J. A. Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain. Physica D: Nonlinear Phenomena, 231,

• Journal Article Type: Article
• Deposit Date: Jul 23, 2008
• Journal: Physica D
• Print ISSN: 0167-2789
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 231
• Keywords: breathers, non-linear waves, discrete systems
• Public URL: https://nottingham-repository.worktribe.com/output/1017242