A Mathematical Model for Transport in Poroelastic Materials with Variable Volume: Derivation, Lie Symmetry Analysis and Examples. Part 2

The model for perfused tissue undergoing deformation taking into account the local exchange between tissue and blood and lymphatic systems is presented. The Lie symmetry analysis in order to identify its symmetry properties is applied. Several families of steady-state solutions in closed formulae are derived. An analysis of the impact of the parameter values and boundary conditions on the distribution of hydrostatic pressure, osmotic agent concentration and deformation of perfused tissue is provided applying the solutions obtained in examples describing real-world processes.

Dynamics in a diffusive predator–prey system with double Allee effect and modified Leslie–Gower scheme

In this paper, we investigate a diffusive modified Leslie–Gower predator–prey system with double Allee effect on prey. The global existence, uniqueness and a priori bound of positive solutions are determined. The existence and local stability of constant steady–state solutions are analyzed. Next, we induce the nonexistence of nonconstant positive steady–state solutions, which indicates the effect of large diffusivity. Furthermore, we discuss the steady–state bifurcation and the existence of nonconstant positive steady–state solutions by the bifurcation theory. In addition, Hopf bifurcations of the spatially homogeneous and inhomogeneous periodic orbits are studied. Finally, we make some numerical simulations to validate and complement the theoretical analysis. Our results demonstrate that the dynamics of the system with double Allee effect and modified Leslie–Gower scheme are richer and more complex.

On the Dynamics of Low-viscosity Warped Disks around Black Holes

Accretion disks around black holes can become warped by Lense–Thirring precession. When the disk viscosity is sufficiently small, such that the warp propagates as a wave, then steady-state solutions to the linearized fluid equations exhibit an oscillatory radial profile of the disk tilt angle. Here we show, for the first time, that these solutions are in good agreement with three-dimensional hydrodynamical simulations, in which the viscosity is isotropic and measured to be small compared to the disk angular semi-thickness, and in the case that the disk tilt—and thus the warp amplitude—remains small. We show, using both the linearized fluid equations and hydrodynamical simulations, that the inner disk tilt can be more than several times larger than the original disk tilt, and we provide physical reasoning for this effect. We explore the transition in disk behavior as the misalignment angle is increased, finding increased dissipation associated with regions of strong warping. For large enough misalignments the disk becomes unstable to disk tearing and breaks into discrete planes. For the simulations we present here, we show that the total (physical and numerical) viscosity at the time the disk breaks is small enough that the disk tearing occurs in the wave-like regime, substantiating that disk tearing is possible in this region of parameter space. Our simulations demonstrate that high spatial resolution, and thus low numerical viscosity, is required to accurately model the warp dynamics in this regime. Finally, we discuss the observational implications of our results.

Stability features of steady-state solutions for a diode with electron and ion counter-streams

Stability features of steady-state solutions for a diode with counter-streaming electron and ion flows are studied. For this purpose, the time-dependent problem for an exponential potential perturbation with complex frequency is considered. By linearization of the Poisson equation and electron and ion densities integrodifferential equation for the potential perturbation amplitude is derived. In the case of uniform unperturbed potential distribution an explicit solution of this equation is obtained. Eigen modes of the perturbation are studied. The limiting value of the diode length above which steady state solutions in question are unstable is found. The obtained analytical Eigen modes coincide with the result of numerical simulation of the potential perturbation evolution.

Application of Two-dimensional Finite Volume Method to Protoplanetary Disks

Many fascinating astrophysical phenomena can be simulated insufficiently by standard numerical schemes for the compressible hydrodynamics equations. In the present work, a high performant 2D hydrodynamical code has been developed. The model is designed for the planetary formation that consists of momentum, continuity and energy equations. Since the two-phase model seems to be hardly executed, we will show in a simplified form, the implementation of this model in one-phase. It is applied to the Solar System that such stars can form planets. The finite volume method (FVM) is used in this model. We aim to develop a first-order well-balanced scheme for the Euler equations in the the radial direction, combined with second-order centered ux following the radial direction. This conception is devoted to balance the uxes, and guarantee hydrostatic equilibrium preserving. Then the model is used on simplified examples in order to show its ca- pability to maintain steady-state solutions with a good precision. Additionally, we demonstrate the performance of the numerical code through simulations. In particularly, the time evolution of gas orbited around the star, and some proper- ties of the Rossby wave instability are analyzed. The resulting scheme shows consequently that this model is robust and simple enough to be easily implemented.

Stabilizing nanolasers via polarization lifetime tuning

We investigate the emission dynamics of mutually coupled nanolasers and predict ways to optimize their stability, i.e., maximize their locking range. We find that tuning the cavity lifetime to the same order of magnitude as the dephasing time of the microscopic polarization yields optimal operation conditions, which allow for wider tuning ranges than usually observed in conventional semiconductor lasers. The lasers are modeled by Maxwell–Bloch type class-C equations. For our analysis, we analytically determine the steady state solutions, analyze the symmetries of the system and numerically characterize the emission dynamics via the underlying bifurcation structure. The polarization lifetime is found to be a crucial parameter, which impacts the observed dynamics in the parameter space spanned by frequency detuning, coupling strength and coupling phase.

Steady-State Problem of an S-K-T Competition Model with Spatially Degenerate Coefficients

In this paper, we study the steady-state problem of an S-K-T competition model with a spatially degenerate intraspecific competition coefficient. First, the global bifurcation continuum of positive steady-state solutions from its semitrivial steady-state solution is given, which depends on the spatial heterogeneity and cross-diffusion. Second, two limiting systems are derived as the cross-diffusion coefficient tends to infinity. Moreover, we demonstrate the existence of positive steady-state solutions near the two limiting systems, and show which one of the limiting systems characterizes the positive steady-state solution.

Uncertain Circuit Equation

Differential equation is a powerful tool for investigating the transient and steady-state solutions of electrical circuit in the time domain. By considering the noise in actual circuit system, this paper first presents an uncertain circuit equation, which is a type of differential equation driven by Liu process. Then the solution of uncertain circuit equation and the inverse uncertainty distribution of solution are derived. Following that, two applications of solution are provided as well. Based on the observations, the method of moments is used to estimate the unknown parameters in uncertain circuit equation. In addition, a paradox for stochastic circuit equation is also given.