Discrete Breathers in One- and Two-Dimensional Lattices

Butt, Imran Ashiq (2006) Discrete Breathers in One- and Two-Dimensional Lattices. PhD thesis, University of Nottingham.

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Discrete breathers are time-periodic and spatially localised exact

solutions in translationally invariant nonlinear lattices. They

are generic solutions, since only moderate conditions are required

for their existence. Closed analytic forms for breather solutions

are generally not known. We use asymptotic methods to determine

both the properties and the approximate form of discrete breather

solutions in various lattices.

We find the conditions for which the one-dimensional FPU chain

admits breather solutions, generalising a known result for

stationary breathers to include moving breathers. These

conditions are verified by numerical simulations. We show that the

FPU chain with quartic interaction potential supports long-lived

waveforms which are combinations of a breather and a kink. The

amplitude of classical monotone kinks is shown to have a nonzero

minimum, whereas the amplitude of breathing-kinks can be

arbitrarily small.

We consider a two-dimensional FPU lattice with square rotational

symmetry. An analysis to third-order in the wave amplitude is

inadequate, since this leads to a partial differential equation

which does not admit stable soliton solutions for the breather

envelope. We overcome this by extending the analysis to

higher-order, obtaining a modified partial differential equation

which includes known stabilising terms. From this, we determine

regions of parameter space where breather solutions are expected.

Our analytic results are supported by extensive numerical

simulations, which suggest that the two-dimensional square FPU

lattice supports long-lived stationary and moving breather modes.

We find no restriction upon the direction in which breathers can

travel through the lattice. Asymptotic estimates for the breather

energy confirm that there is a minimum threshold energy which must

be exceeded for breathers to exist in the two-dimensional lattice.

We find similar results for a two-dimensional FPU lattice with

hexagonal rotational symmetry.

Item Type: Thesis (University of Nottingham only) (PhD)
Supervisors: Wattis, Jonathan A D
Richardson, Giles W
Keywords: discrete breathers, intrinsic localised modes, lattice solitons, localisation, nonlinear waves, discrete systems, continuum approximations
Subjects: Q Science > QA Mathematics > QA150 Algebra
Faculties/Schools: UK Campuses > Faculty of Science > School of Mathematical Sciences
Item ID: 10238
Depositing User: EP, Services
Date Deposited: 24 Apr 2007
Last Modified: 17 Oct 2017 21:23
URI: https://eprints.nottingham.ac.uk/id/eprint/10238

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