Existence, uniqueness and travelling waves to model an invasive specie interaction with heterogeneous reaction and non-linear diffusion

<abstract><p>It is the objective to provide a mathematical treatment of a model to predict the behaviour of an invasive specie proliferating in a domain, but with a certain hostile zone. The behaviour of the invasive is modelled in the frame of a non-linear diffusion (of Porous Medium type) equation with non-Lipschitz and heterogeneous reaction. First of all, the paper examines the existence and uniqueness of solutions together with a comparison principle. Once the regularity principles are shown, the solutions are studied within the Travelling Waves (TW) domain together with stability analysis in the frame of the Geometric Perturbation Theory (GPT). As a remarkable finding, the obtained TW profile follows a potential law in the stable connection that converges to the stationary solution. Such potential law suggests that the pressure induced by the invasive over the hostile area increases over time. Nonetheless, the finite speed, induced by the non-linear diffusion, slows down a possible violent invasion.</p></abstract>

Long-wave equation for a confined ferrofluid interface: periodic interfacial waves as dissipative solitons

We study the dynamics of a ferrofluid thin film confined in a Hele-Shaw cell, and subjected to a tilted non-uniform magnetic field. It is shown that the interface between the ferrofluid and an inviscid outer fluid (air) supports travelling waves, governed by a novel modified Kuramoto–Sivashinsky-type equation derived under the long-wave approximation. The balance between energy production and dissipation in this long-wave equation allows for the existence of dissipative solitons. These permanent travelling waves’ propagation velocity and profile shape are shown to be tunable via the external magnetic field. A multiple-scale analysis is performed to obtain the correction to the linear prediction of the propagation velocity, and to reveal how the nonlinearity arrests the linear instability. The travelling periodic interfacial waves discovered are identified as fixed points in an energy phase plane. It is shown that transitions between states (wave profiles) occur. These transitions are explained via the spectral stability of the travelling waves. Interestingly, multi-periodic waves, which are a non-integrable analogue of the double cnoidal wave, are also found to propagate under the model long-wave equation. These multi-periodic solutions are investigated numerically, and they are found to be long-lived transients, but ultimately abruptly transition to one of the stable periodic states identified.

COMSOL modeling of the lightning-generated electric field distribution in a chain of insulators of a 115 kV power transmission line

The transmission lines installed in our city, to the outskirts of it and in the rest of the national territory produce a level of emissions of electric field, these levels vary depending on the physical disposition of the drivers, the distance between spans, the voltage level of the line, among others. These values must respect the exposure limits for individuals, as stipulated in Article 14 of RETIE. In this work will be developed a study of distribution of the electric field along a chain of insulators standard of tempered glass of a transmission line of 115 kV with the aid of the software COMSOL Multiphysics, to the impact of an atmospheric discharge on the storage cable with the help of ATPDraw software, taking into account variations in the value of the earthing resistance of the structure. With the electric field distribution values obtained by means of simulations carried out in the COMSOL Multiphysics software, comparisons are made of the magnitude of the obtained field, taking as a reference the stable state of the system without condition of pollutants. Initially a simulation of a transmission line of 115 kV is performed in the ATPDraw software, analyzing the lightning impulse generated on the guard cable, at different values of grounding resistance, to determine the waveform and reached value of overvoltage in the insulator chain. Then the overvoltage signal obtained with the help of ATPDraw software is recreated in the COMSOL Multiphysics software, to be able to visualize the distribution and behavior of the electric field along the chain, taking into account the different factors involved in the process, pollutants such as salinity, themes such as travelling waves etc. and thus determine possible line isolation failures by comparing the values achieved by simulating in COMSOL Multiphysics with respect to the CFO of the insulator chain.

Nonlinear system of PT-symmetric excitations and Toda vibrations integrable by the Darboux–Bäcklund dressing method

The nonlinear dynamics of coupled P T -symmetric excitations and Toda-like vibrations on a one-dimensional lattice are studied analytically and elucidated graphically. The nonlinear exciton-phonon system as the whole is shown to be integrable in the Lax sense inasmuch as it admits the zero-curvature representation supported by the auxiliary linear problem of third order. Inspired by this fact, we have developed in detail the Darboux–Bäcklund integration technique appropriate to generate a higher-rank crop solution by dressing a lower-rank (supposedly known) seed solution. In the framework of this approach, we have found a rather non-trivial four-component analytical solution exhibiting the crossover between the monopole and dipole regimes in the spatial distribution of intra-site excitations. This effect is inseparable from the pronounced mutual influence between the interacting subsystems in the form of specific nonlinear superposition of two essentially distinct types of travelling waves. We have established the criterion of monopole-dipole transition based upon the interplay between the localization parameter of Toda mode and the inter-subsystem coupling parameter.

Influence of nozzle external geometry on wavepackets in under-expanded supersonic impinging jets

In this study, large-eddy simulations are utilised to unravel the influence of the nozzle's external geometry on upstream-travelling waves in under-expanded supersonic impinging jets. Three configurations, a thin-lipped, a thin-lipped with a sponge and an infinite-lipped nozzle are considered with the other non-dimensionalised geometrical and flow variables identical for the three cases. Spectral proper orthogonal decomposition is applied to the Mack norm, i.e. the energy norm based on the stagnation energy, to obtain the spatial modes at their corresponding frequency. The spectral decomposition of the spatial modes at optimal and suboptimal frequencies is used to isolate the wavepackets into upstream- and downstream-propagating waves based on their phase velocity. It is found that the external geometry of the nozzle has a significant influence on the first-order statistics even though the governing non-dimensional parameters are the same for all three cases. Multiple peaks emerge in the energy spectra at distinct frequencies corresponding to axisymmetric azimuthal modes for each case. The downstream-propagating wavepackets have a high amplitude at the shear layer of the three jets with the mode shapes resembling Kelvin–Helmholtz instability waves, while the upstream-travelling wavepackets exist in the three regions of the near field, shear layer and inside of the jet. The barrel shock at the nozzle exit appears as a flexible shield, which prevents upstream-travelling waves from reaching the internal region of the nozzle, where the upstream-travelling waves travel obliquely with one side of the wavefront is crawling on the reflected shock while the other side is guided by the shear layer. These latter waves can reach the nozzle lip via inside of the jet. The spectral decomposition of the spatial modes at optimal and suboptimal frequencies show that all three forms of the near field, shear layer and inside jet upstream-travelling wavepackets contribute to the receptivity process while their contributions and strength are altered by the change of the external geometry of the nozzle.

The ascending arousal system shapes neural dynamics to mediate awareness of cognitive states

Models of cognitive function typically focus on the cerebral cortex and hence overlook functional links to subcortical structures. This view does not consider the role of the highly-conserved ascending arousal system’s role and the computational capacities it provides the brain. We test the hypothesis that the ascending arousal system modulates cortical neural gain to alter the low-dimensional energy landscape of cortical dynamics. Here we use spontaneous functional magnetic resonance imaging data to study phasic bursts in both locus coeruleus and basal forebrain, demonstrating precise time-locked relationships between brainstem activity, low-dimensional energy landscapes, network topology, and spatiotemporal travelling waves. We extend our analysis to a cohort of experienced meditators and demonstrate locus coeruleus-mediated network dynamics were associated with internal shifts in conscious awareness. Together, these results present a view of brain organization that highlights the ascending arousal system’s role in shaping both the dynamics of the cerebral cortex and conscious awareness.