Moving Embedded Solitons

The first theoretical results are reported predicting {\em moving} solitons residing inside ({\it embedded} into) the continuous spectrum of radiation modes. The model taken is a Bragg-grating medium with Kerr nonlinearity and additional second-derivative (wave) terms. The moving embedded solitons (ESs) are doubly isolated (of codimension 2), but, nevertheless, structurally stable. Like quiescent ESs, moving ESs are argued to be stable to linear approximation, and {\it semi}-stable nonlinearly. Estimates show that moving ESs may be experimentally observed as $\sim$10 fs pulses with velocity $\leq 1/10$th that of light.Comment: 9 pages 2 figure

Embedded Solitons in a Three-Wave System

We report a rich spectrum of isolated solitons residing inside ({\it embedded } into) the continuous radiation spectrum in a simple model of three-wave spatial interaction in a second-harmonic-generating planar optical waveguide equipped with a quasi-one-dimensional Bragg grating. An infinite sequence of fundamental embedded solitons are found, which differ by the number of internal oscillations. Branches of these zero-walkoff spatial solitons give rise, through bifurcations, to several secondary branches of walking solitons. The structure of the bifurcating branches suggests a multistable configuration of spatial optical solitons, which may find straightforward applications for all-optical switching.Comment: 5 pages 5 figures. To appear in Phys Rev

Factors affecting distribution and habitat selection of water shrews Neomys fodiens

The water shrew Neomys fodiens is one of Britain’s least known mammals and its habitat requirements are poorly understood. The purpose of this study was to determine occurrence and associated habitat preferences of water shrews, a species of conservation concern, by comparing populations in central England freshwater habitats. Bait tube surveys were undertaken at 32 freshwater sites to establish water shrew presence, half of which were found to contain water shrews. Habitat surveys were undertaken and, in addition to water shrew presence/absence data, were used to develop habitat suitability index models by means of artificial neural networks. Management intensity (occasional or frequent bankside management) was identified as the most important predictor of water shrew presence and, when combined with dissolved oxygen (0-2.99mg l-1) and water depth (<25cm), created the highest performing model. These models will allow sites to be rapidly assessed for water shrew presence without labour intensive and costly live-trapping techniques. Prey availability was investigated by undertaking invertebrate surveys at four water shrew-positive sites, as well as at an additional four sites with unknown water shrew presence with which to compare

Thirring Solitons in the presence of dispersion

The effect of dispersion or diffraction on zero-velocity solitons is studied for the generalized massive Thirring model describing a nonlinear optical fiber with grating or parallel-coupled planar waveguides with misaligned axes. The Thirring solitons existing at zero dispersion/diffraction are shown numerically to be separated by a finite gap from three isolated soliton branches. Inside the gap, there is an infinity of multi-soliton branches. Thus, the Thirring solitons are structurally unstable. In another parameter region (far from the Thirring limit), solitons exist everywhere.Comment: 12 pages, Latex. To appear in Phys. Rev. Let

Embedded Solitons in Lagrangian and Semi-Lagrangian Systems

We develop the technique of the variational approximation for solitons in two directions. First, one may have a physical model which does not admit the usual Lagrangian representation, as some terms can be discarded for various reasons. For instance, the second-harmonic-generation (SHG) model considered here, which includes the Kerr nonlinearity, lacks the usual Lagrangian representation if one ignores the Kerr nonlinearity of the second harmonic, as compared to that of the fundamental. However, we show that, with a natural modification, one may still apply the variational approximation (VA) to those seemingly flawed systems as efficiently as it applies to their fully Lagrangian counterparts. We call such models, that do not admit the usual Lagrangian representation, \textit{semi-Lagrangian} systems. Second, we show that, upon adding an infinitesimal tail that does not vanish at infinity, to a usual soliton ansatz, one can obtain an analytical criterion which (within the framework of VA) gives a condition for finding \textit{embedded solitons}, i.e., isolated truly localized solutions existing inside the continuous spectrum of the radiation modes. The criterion takes a form of orthogonality of the radiation mode in the infinite tail to the soliton core. To test the criterion, we have applied it to both the semi-Lagrangian truncated version of the SHG model and to the same model in its full form. In the former case, the criterion (combined with VA for the soliton proper) yields an \emph{exact} solution for the embedded soliton. In the latter case, the criterion selects the embedded soliton with a relative error $\approx 1$.Comment: 10 pages, 1 figur

Radiationless Travelling Waves In Saturable Nonlinear Schr\"odinger Lattices

The longstanding problem of moving discrete solitary waves in nonlinear Schr{\"o}dinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. {\em Genuinely localised travelling waves} are computed as a function of the system parameters {\it for the first time}. The relevant solutions exist only for finite velocities.Comment: 5 pages, 4 figure

Origin of Multikinks in Dispersive Nonlinear Systems

We develop {\em the first analytical theory of multikinks} for strongly {\em dispersive nonlinear systems}, considering the examples of the weakly discrete sine-Gordon model and the generalized Frenkel-Kontorova model with a piecewise parabolic potential. We reveal that there are no $2\pi$-kinks for this model, but there exist {\em discrete sets} of $2\pi N$-kinks for all N>1. We also show their bifurcation structure in driven damped systems.Comment: 4 pages 5 figures. To appear in Phys Rev

Stripe to spot transition in a plant root hair initiation model

A generalised Schnakenberg reaction-diffusion system with source and loss terms and a spatially dependent coefficient of the nonlinear term is studied both numerically and analytically in two spatial dimensions. The system has been proposed as a model of hair initiation in the epidermal cells of plant roots. Specifically the model captures the kinetics of a small G-protein ROP, which can occur in active and inactive forms, and whose activation is believed to be mediated by a gradient of the plant hormone auxin. Here the model is made more realistic with the inclusion of a transverse co-ordinate. Localised stripe-like solutions of active ROP occur for high enough total auxin concentration and lie on a complex bifurcation diagram of single and multi-pulse solutions. Transverse stability computations, confirmed by numerical simulation show that, apart from a boundary stripe, these 1D solutions typically undergo a transverse instability into spots. The spots so formed typically drift and undergo secondary instabilities such as spot replication. A novel 2D numerical continuation analysis is performed that shows the various stable hybrid spot-like states can coexist. The parameter values studied lead to a natural singularly perturbed, so-called semi-strong interaction regime. This scaling enables an analytical explanation of the initial instability, by describing the dispersion relation of a certain non-local eigenvalue problem. The analytical results are found to agree favourably with the numerics. Possible biological implications of the results are discussed.Comment: 28 pages, 44 figure