An explanation is offered for an observed lower bound on the wave speed of travelling kinks in Frenkel-Kontorova lattices. Kinks exist at discrete wavespeeds within a parameter regime where there is a resonance with linear waves (phonons). However, they fail to exist even in this codimension-one sense whenever there is more than one phonon branch in the dispersion relation; inside such bands only quasi-kinks with nondecaying oscillatory tails are possible. The results are presented for a discrete sine-Gordon lattice with an onsite potential that has a tunable amount of anharmonicity. Novel numerical methods are used to trace kinks with topological charge Q = 1 and 2 in three parameters representing the propagation speed, lattice discreteness and anharmonicity. Although none of the analysis is presented as rigorous mathematics, numerical results suggest that the bound on allowable wavespeeds is sharp. The results also explain why the vanishing, at discrete values of anharmonicity, of the Peierls-Nabarro barrier between stationary kinks as discovered by Savin et al., does not lead to bifurcation of kinks with small wave speed.

A variational model is formulated that accounts for the localization of deformation due to buckling under pure bending of thin-walled elastic tubes with circular cross-sections. Previous studies have successfully modelled the gradual process of ovalization of the cross-section with an accompanying progressive reduction in stiffness but these theories have had insufficient freedom to incorporate any longitudinal variation in the tube. Here, energy methods and small-strain nonlinear elastic theory are used to model the combined effects of cross-section deformation and localized longitudinal buckling. Results are compared with a number of case studies, including a nanotube, and it is found that the model gives rise to behaviours that correlate well with some published physical experiments and numerical studies.

This paper deals with the numerical veriÿcation of the theory developed by Derzho and Grimshaw (DG) (1997, Phys. Fluids 9(11), 3378-3385) regarding solitary waves in stratiÿed uids with recirculation regions. The Boussinesq approximation is made and the stratiÿcation is chosen such that the Brunt-V ais al a frequency di ers only slightly from uniform stratiÿcation. To establish the consistency of the numerical scheme the usual KdV and mKdV solutions are tested ÿrst and then the solutions obtained by DG are considered. It is found that these waves remain of permanent form and are stationary when viewed at their corresponding phase speed. The recirculation region remains stagnant to ÿrst order as predicted by DG.

A high-resolution spectral numerical scheme is developed to solve the two-dimensional equations of motion for the ow of a density stratiÿed, incompressible and inviscid uid. This method incorporates the inertial terms neglected in the Boussinesq approximation. Thus it aims, inter alia, to extend the numerical simulations of Rottman et al. (J. Fluid Mech. 306 (1996) 1) and Aigner et al. (Fluid Dyn. 25 (1999) 315). The validity of the numerical model is tested with two applications. The ÿrst application is the resonant ow over isolated bottom topography in a channel of ÿnite depth, which has been studied extensively in the Boussinesq approximation. The inclusion of inertial e ects, that is the in uence of the stratiÿcation on the acceleration terms discarded in the Boussinesq approximation, allows the comparison of the solution to the unsteady governing equations with the fully nonlinear, but weakly dispersive resonant theory of Grimshaw and Yi (J. Fluid Mech. 229 (1991) 603). This paper focuses on topography of small-to-moderate amplitudes and slopes, and for conditions such that the ow is close to linear resonance for the ÿrst internal wave mode. The vertical position of wave breaking is determined. The second application is the propagation of large-amplitude internal solitary waves with vortex cores, again in a channel of ÿnite depth. The existence and permanence of these types of waves derived by Derzho and Grimshaw (Phys. Fluids 9(11) (1997) 3378) is veriÿed. Furthermore, the time-dependent solution provides measurements of the structure of the vortex core and maximum adverse velocity at the top boundary.