Travelling waves arise in several areas of science, hence modification of travelling wave properties is of great interest. While many studies have demonstrated how to control the form or shape of a solitary travelling wave by employing soliton or dis
persion management, far less is known about controlling the motion of a travelling wave while keeping its form unchanged. We present a technique for control of travelling wave motion using time-varying coefficients, which we refer to as wave management. The technique allows one to alter the trajectory of a travelling wave, slowing, stopping, or reversing the direction of the wave, all while ensuring that the wave form is unchanged, and we illustrate this through multiple examples. Our results suggest that wave management is a promising tool for applications where one needs to modify the motion of a wave while preserving its form, and we highlight several potential applications.
We study coupled unstaggered-staggered soliton pairs emergent from a system of two coupled discrete nonlinear Schr{o}dinger (DNLS) equations with the self-attractive on-site self-phase-modulation nonlinearity, coupled by the repulsive cross-phase-mod
ulation interaction, on 1D and 2D lattice domains. These mixed modes are of a symbiotic type, as each component in isolation may only carry ordinary unstaggered solitons. While most work on DNLS systems addressed symmetric on-site-centered fundamental solitons, these models give rise to a variety of other excited states, which may also be stable. The simplest among them are antisymmetric states in the form of discrete twisted solitons, which have no counterparts in the continuum limit. In the extension to 2D lattice domains, a natural counterpart of the twisted states are vortical solitons. We first introduce a variational approximation (VA) for the solitons, and then correct it numerically to construct exact stationary solutions, which are then used as initial conditions for simulations to check if the stationary states persist under time evolution. Two-component solutions obtained include (i) 1D fundamental-twisted and twisted-twisted soliton pairs, (ii) 2D fundamental-fundamental soliton pairs, and (iii) 2D vortical-vortical soliton pairs. We also highlight a variety of other transient dynamical regimes, such as breathers and amplitude death. The findings apply to modeling binary Bose-Einstein condensates, loaded in a deep lattice potential, with identical or different atomic masses of the two components, and arrays of bimodal optical waveguides.
The study of pattern-forming instabilities in reaction-diffusion systems on growing or otherwise time-dependent domains arises in a variety of settings, including applications in developmental biology, spatial ecology, and experimental chemistry. Ana
lyzing such instabilities is complicated, as there is a strong dependence of any spatially homogeneous base states on time, and the resulting structure of the linearized perturbations used to determine the onset of instability is inherently non-autonomous. We obtain general conditions for the onset and structure of diffusion driven instabilities in reaction-diffusion systems on domains which evolve in time, in terms of the time-evolution of the Laplace-Beltrami spectrum for the domain and functions which specify the domain evolution. Our results give sufficient conditions for diffusive instabilities phrased in terms of differential inequalities which are both versatile and straightforward to implement, despite the generality of the studied problem. These conditions generalize a large number of results known in the literature, such as the algebraic inequalities commonly used as a sufficient criterion for the Turing instability on static domains, and approximate asymptotic results valid for specific types of growth, or specific domains. We demonstrate our general Turing conditions on a variety of domains with different evolution laws, and in particular show how insight can be gained even when the domain changes rapidly in time, or when the homogeneous state is oscillatory, such as in the case of Turing-Hopf instabilities. Extensions to higher-order spatial systems are also included as a way of demonstrating the generality of the approach.
There are several models for the effective thermal conductivity of two-phase composite materials in terms of the conductivity of the solid and the disperse material. In this paper, we generalise three models of Maxwell type (namely, the classical Max
well model and two generalisations of it obtained from effective medium theory and differential effective medium theory) so that the resulting effective thermal conductivity accounts for radiative heat transfer within gas voids. In the high-temperature regime, radiative transfer within voids strongly influences the thermal conductivity of the bulk material. Indeed, the utility of these models over classical Maxwell-type models is seen in the high-temperature regime, where they predict that the effective thermal conductivity of the composite material levels off to a constant value (as a function of temperature) at very high temperatures, provided that the material is not too porous, in agreement with experiments. This behaviour is in contrast to models which neglect radiative transfer within the pores, or lumped parameter models, as such models do not resolve the radiative transfer independently from other physical phenomena. Our results may be of particular use for industrial and scientific applications involving heat transfer within porous composite materials taking place in the high-temperature regime.
We study wave propagation in two-dimensional granular crystals under the Hertzian contact law consisting of hexagonal packings of spheres under various basin geometries including hexagonal, triangular, and circular basins which can be tiled with hexa
gons. We find that the basin geometry will influence wave reflection at the boundaries, as expected, and also may result in bottlenecks forming. While exterior strikers the size of a single sphere have been considered in the literature, it is also possible to consider strikers which impact multiple spheres along a boundary, or to have multiple sides being struck simultaneously. It is also possible to consider obstructions or even strikers in the interior of the hexagonally packed granular crystal, as previously considered in the case of square packings, resulting in the basin geometry no longer forming a convex set. We consider various configurations of either boundary or interior strikers. We shall also consider the case where a granular crystal is composed of two separate crystals of differing material, with a single interface between the two distinct materials. Depending on the relative material properties of each type of sphere, this can result in a trapping of most of the wave energy within one of the two regions. While repeated reflections from the boundaries will cause the systems we study to fall into disorder for large time, there are a number of interesting wave structures and patters that emerge as transients at intermediate timescales.
Amplitude death, which occurs in a system when one or more macroscopic wavefunctions collapse to zero, has been observed in mutually coupled solid-state lasers, analog circuits, and thermoacoustic oscillators, to name a few applications. While studie
s have considered amplitude death on oscillator systems and in externally forced complex Ginzburg-Landau systems, a route to amplitude death has not been studied in autonomous continuum systems. We derive simple analytic conditions for the onset of amplitude death of one macroscopic wavefunction in a system of two coupled complex Ginzburg-Landau equations with general nonlinear self- and cross-interaction terms. Our results give a more general theoretical underpinning for recent amplitude death results reported in the literature, and suggest an approach for tuning parameters in such systems so that they either permit or prohibit amplitude death of a wavefunction (depending on the application). Numerical simulation of the coupled complex Ginzburg-Landau equations, for examples including cubic, cubic-quintic, and saturable nonlinearities, is used to illustrate the analytical results.
We formulate and study dynamics from a complex Ginzburg-Landau system with saturable nonlinearity, including asymmetric cross-phase modulation (XPM) parameters. Such equations can model phenomena described by complex Ginzburg-Landau systems under the
added assumption of saturable media. When the saturation parameter is set to zero, we recover a general complex cubic Ginzburg-Landau system with XPM. We first derive conditions for the existence of bounded dynamics, approximating the absorbing set for solutions. We use this to then determine conditions for amplitude death of a single wavefunction. We also construct exact plane wave solutions, and determine conditions for their modulational instability. In a degenerate limit where dispersion and nonlinearity balance, we reduce our system to a saturable nonlinear Schrodinger system with XPM parameters, and we demonstrate the existence and behavior of spatially heterogeneous stationary solutions in this limit. Using numerical simulations we verify the aforementioned analytical results, while also demonstrating other interesting emergent features of the dynamics, such as spatiotemporal chaos in the presence of modulational instability. In other regimes, coherent patterns including uniform states or banded structures arise, corresponding to certain stable stationary states. For sufficiently large yet equal XPM parameters, we observe a segregation of wavefunctions into different regions of the spatial domain, while when XPM parameters are large and take different values, one wavefunction may decay to zero in finite time over the spatial domain (in agreement with the amplitude death predicted analytically). While saturation will often regularize the dynamics, such transient dynamics can still be observed - and in some cases even prolonged - as the saturability of the media is increased, as the saturation may act to slow the timescale.
We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in
a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.
A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue en
gineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport, and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold, and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models, and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.
We study solitary wave propagation in 1D granular crystals with Hertz-like interaction potentials. We consider interfaces between media with different exponents in the interaction potential. For an interface with increasing interaction potential expo
nent along the propagation direction we obtain mainly transmission with delayed secondary transmitted and reflected pulses. For interfaces with decreasing interaction potential exponent we observe both significant reflection and transmission of the solitary wave, where the transmitted part of the wave forms a multipulse structure. We also investigate impurities consisting of beads with different interaction exponents compared to the media they are embedded in, and we find that the impurities cause both reflection and transmission, including the formation of multipulse structures, independent of whether the exponent in the impurities is smaller than in the surrounding media. We explain wave propagation effects at interfaces and impurities in terms of quasi-particle collisions. Next we consider wave propagation along Hertz-like granular chains of beads in the presence of disorder and periodicity in the interaction exponents present in the Hertz-like potential, modelling, for instance, inhomogeneity in the contact geometry between beads in the granular chain. We find that solitary waves in media with randomised interaction exponents (which models disorder in the contact geometry) experience exponential decay, where the dependence of the decay rate is similar to the case of randomised bead masses. In the periodic case of chains with interaction exponents alternating between two fixed values, we find qualitatively different propagation properties depending on the choice of the two exponents. In particular, we find regimes with either exponential decay or stable solitary wave propagation with pairwise collective behaviour.
