Butt, Imran Ashiq (2006) Discrete Breathers in One- and Two-Dimensional Lattices. PhD thesis, University of Nottingham.
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
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 (PhD)|
|Supervisors:||Wattis, Jonathan A D|
Richardson, Giles W
|Uncontrolled Keywords:||discrete breathers, intrinsic localised modes, lattice solitons, localisation, nonlinear waves, discrete systems, continuum approximations|
|Faculties/Schools:||UK Campuses > Faculty of Science > School of Mathematical Sciences|
|Deposited By:||Imran Ashiq Butt|
|Deposited On:||24 Apr 2007|
|Last Modified:||06 Feb 2009 14:43|
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