Jonathan A. D. Wattis

Existence theorem for solitary waves on lattices

AbstractIn this article we give an existence theorem for localized travelling wave solutions on one-dimensional lattices with Hamiltonian

Symmetry-breaking in chiral polymerisation

H = \sum\limits_{n \in \mathbb{Z}} {\left( {\tfrac{1}{2}p_n^2 + V(q_{n + 1} - q_n )} \right)} ,

Quasi-continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation

whereV(·) is the potential energy due to nearest-neighbour interactions. Until now, apart from rare integrable lattices like the Toda latticeV(φ)=ab−1(e−bφ+bφ−1), the only evidence for existence of such solutions has been numerical. Our result in particular recovers existence of solitary waves in the Toda lattice, establishes for the first time existence of solitary waves in the (nonintegrable) cubic and quartic latticesV(φ)= 1/2φ2 + 1/3aφ3,V(φ) = 1/2φ2 + 1/4bφ4, thereby confirming the numerical findings in [1] and shedding new light on the recurrence phenomena in these systems observed first by Fermi, Pasta and Ulam [2], and shows that contrary to widespread belief, the presence of exact solitary waves is not a peculiarity of integrable systems, but “generic” in this class of nonlinear lattices. The approach presented here is new and quite general, and should also be applicable to other forms of lattice equations: the travelling waves are sought as minimisers of a naturally associated variational problem (obtained via Hamiltons principle), and existence of minimisers is then established using modern methods in the calculus of variations (the concentration-compactness principle of P.-L. Lions [3]).

Analysis of a Generalized Becker-Doring Model of Self-Reproducing Micelles

Abstract We examine a variety of numerical and approximate analytical methods to study families of solitary waves on lattices. Such waves, when they exist, travel through the lattice without loss of energy, and have approximate soliton properties on collision. Corresponding quantum problems are also briefly described.

Approximations to solitary waves on lattices. II. quasi-continuum methods for fast and slow waves

We propose a model for chiral polymerisation and investigate its symmetric and asymmetric solutions. The model has a source species which decays into left- and right-handed types of monomer, each of which can polymerise to form homochiral chains; these chains are susceptible to ‘poisoning’ by the opposite-handed monomer. Homochiral polymers are assumed to influence the proportion of each type of monomer formed from the precursor. We show that for certain parameter values a positive feedback mechanism makes the symmetric steady-state solution unstable.The kinetics of polymer formation are then analysed in the case where the system starts from zero concentrations of monomers and chains. We show that following a long induction time, extremely large concentrations of polymers are formed for a short time, during this time an asymmetry introduced into the system by a random external perturbation may be massively amplified. The system then approaches one of the steady-state solutions described above.

Mathematical Models of the Homochiralisation of Crystals by Grinding

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 Schrodinger (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

Flow dynamics in a stented ureter.

(2+1)

Asymptotic solutions of the Becker-Döring equations

-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.