Research Repository

# Discrete breathers in a two-dimensional Fermi-Pasta-Ulam lattice

## Authors

Imran A. Butt

### Abstract

Using asymptotic methods, we investigate whether discrete
breathers are supported by a two-dimensional Fermi-Pasta-Ulam lattice. A scalar (one-component) two-dimensional
Fermi-Pasta-Ulam lattice is shown to model the charge stored
within an electrical transmission lattice. A third-order multiple-scale analysis in the semi-discrete limit fails, since at this order, the lattice equations reduce to the (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation which does not support stable soliton solutions for the breather envelope. We therefore extend
the analysis to higher order and find a generalised
$(2+1)$-dimensional NLS equation which incorporates higher order dispersive and nonlinear terms as perturbations. We find an ellipticity criterion for the wave numbers of the carrier wave. Numerical simulations suggest that both stationary and moving breathers are supported by the system. Calculations of the energy show the expected threshold behaviour whereby the energy of breathers does {\em not} go to zero with the amplitude; we find
that the energy threshold is maximised by stationary breathers, and becomes arbitrarily small as the boundary of the domain of ellipticity is approached.

### Citation

Butt, I. A., & Wattis, J. A. Discrete breathers in a two-dimensional Fermi-Pasta-Ulam lattice. Journal of Physics A: Mathematical and General, 39,

Journal Article Type Article Aug 14, 2008 Journal of Physics. A, Mathematical and General 0305-4470 IOP Publishing Peer Reviewed 39 https://nottingham-repository.worktribe.com/output/1018711

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