Inhomogeneous Solutions Research Articles

Nonlinear asymptotic stability of inhomogeneous steady solutions to boundary problems of Vlasov–Poisson equation

Nonlinear asymptotic stability of inhomogeneous steady solutions to boundary problems of Vlasov–Poisson equation

Precision cosmology with exact inhomogeneous solutions of general relativity: The Szekeres models

Precision cosmology with exact inhomogeneous solutions of general relativity: The Szekeres models

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Vigneron (Found Phys 54:15, https://doi.org/10.1007/s10701-023-00749-z, 2024) recently proposed a modification of general relativity in which a non-dynamical term related to the spatial topology is introduced in the Einstein equation. The original motivation for this theory is to allow for the non-relativistic limit to exist in any physical topology. In the present paper, we derive a first inhomogeneous exact vacuum solution of this theory for a spherical topology, assuming staticity and spherical symmetry. The metric represents a black hole and a repulsive singularity at opposite poles of a 3-sphere. The solution is similar to the Schwarzschild metric, but the spacelike infinity is cut, and replaced by a repulsive singularity at finite distance, implying that the spacelike hypersurfaces have finite volume, and the total mass is zero. We discuss how this solution paves the way to massive, non-static solutions of this theory, more directly relevant for cosmology.

Multiphase Viscoelastic Non‐Newtonian Fluid Simulation

AbstractWe propose an SPH‐based method for simulating viscoelastic non‐Newtonian fluids within a multiphase framework. For this, we use mixture models to handle component transport and conformation tensor methods to handle the fluid's viscoelastic stresses. In addition, we consider a bonding effects network to handle the impact of microscopic chemical bonds on phase transport. Our method supports the simulation of both steady‐state viscoelastic fluids and discontinuous shear behavior. Compared to previous work on single‐phase viscous non‐Newtonian fluids, our method can capture more complex behavior, including material mixing processes that generate non‐Newtonian fluids. We adopt a uniform set of variables to describe shear thinning, shear thickening, and ordinary Newtonian fluids while automatically calculating local rheology in inhomogeneous solutions. In addition, our method can simulate large viscosity ranges under explicit integration schemes, which typically requires implicit viscosity solvers under earlier single‐phase frameworks.

Turing–Hopf Bifurcation Coinduced by Diffusion and Delay in Gierer–Meinhardt Systems

This paper investigates the impact of gene expression time delay and diffusion on the dynamic behavior of a class of Gierer–Meinhardt systems under Neumann boundary conditions. It provides necessary and sufficient conditions for the emergence of Hopf, Turing, and Turing–Hopf bifurcations. Utilizing the normal form of the Turing–Hopf bifurcation, the spatiotemporal dynamics close to the bifurcation point are categorized into six types, encompassing spatially inhomogeneous and homogeneous periodic solutions, as well as spatially homogeneous and inhomogeneous steady states, along with their transitions. Notably, it’s observed that the systems may lack stable spatially inhomogeneous periodic solutions under specific parameters, despite the emergence of Turing–Hopf bifurcation. These findings are supported by numerical simulations.

Quantum‐safe Lattice‐based mutual authentication and key‐exchange scheme for the smart grid

AbstractThe smart grid network (SGN) is expected to leverage advances in the Internet of Things (IoT) to enable effective delivery and monitoring of energy. By integrating communication, computing, and information tools like smart sensors and meters to facilitate the process of monitoring, predictions, and management of power usage, the SGN can improve competence of power‐grid architecture. However, the effective deployment of IoT‐powered SGNs hinges on the deployment of strong security protocols. With the advent of quantum computers, classic cryptographic algorithms based on integer factorization and the Diffie‐Hellman assumptions may not be suitable to secure the sensitive data of SGNs. Therefore, in this paper, a secure quantum‐safe mutual authentication and key‐exchange (MAKe) mechanism is proposed for SGNs, that make use of the hard assumptions of small integer solution and inhomogeneous small integer solution problems of lattice. The proposed protocol is intended to offer confidentiality, anonymity, and hashed‐based mutual authentication with a key‐exchange agreement. Similarly, this scheme allows creation and validation of the mutual trust among the smart‐meters (SMs) and neighbourhood‐area network gateway over an insecure wireless channel. A random oracle model is then used to perform the formal security analysis of the proposed approach. A thorough formal analysis demonstrates proposed algorithm's ability to withstand various known attacks. The performance analysis shows that the proposed approach outperforms other comparative schemes with respect to at least 22.07% of minimal energy utilization, 51.48% effective storage and communications costs, as well as 76.28% computational costs, and thus suitable for resource‐constrained SGNs.

Construction of an inhomogeneous solution for the problem of torsion of a radially inhomogeneous transversal isotropic cylinder

Using the method of asymptotic integration of the equations of elasticity theory, inhomogeneous solutions are constructed for the problem of torsion of a radially inhomogeneous transversally isotropic cylinder of small thickness. It is considered that the elastic moduli are arbitrary continuous functions of a variable along the radius of the cylinder. An algorithm is given for constructing exact partial solutions of equilibrium equations for special types of loads, the lateral surface of the cylinder is loaded with forces that depend polynomially on the axial coordinate. Based on the solutions obtained, it is possible to evaluate the areas of applicability of applied theories for radially inhomogeneous cylindrical shells.

A domain wall and chiral edge currents in holographic chiral phase transitions

We investigate spatially inhomogeneous solutions in a top-down holographic model: the D3/D7 model which provides a holographic description of the chiral phase transition for a finite external magnetic field, chemical potential, and temperature. We numerically find a domain wall (or kink) solution in the three dimensional space, which incorporates between the chiral symmetry broken phase at the spatial infinity, under the homogeneous sources. Along with the inhomogeneity of the chiral condensate, the charge density is also spatially modulated. The modulated charge density and finite magnetic field lead to the chiral edge current close to the domain wall. We explore the dependences of those profiles on the chemical potential and temperature near the first and second order phase transition points. Our results indicate that the inhomogeneous solutions we found are in good agreement with those obtained by the Ginzburg-Landau theory in the vicinity of the transition points.

Opers on the projective line, Wronskian relations, and the Bethe Ansatz

It is well-known that the spectra of the Gaudin model may be described in terms of solutions of the Bethe Ansatz equations. A conceptual explanation for the appearance of the Bethe Ansatz equations is provided by appropriate G-opers: G-connections on the projective line with extra structure. In fact, solutions of the Bethe Ansatz equations are parameterized by an enhanced version of opers called Miura opers; here, the opers appearing have only regular singularities. Moreover, this geometric approach to the spectra of the Gaudin model provides a well-known example of the geometric Langlands correspondence. Feigin, Frenkel, Rybnikov, and Toledano Laredo have introduced an inhomogeneous version of the Gaudin model; this model incorporates an additional twist factor, which is an element of the Lie algebra of G. They exhibited the Bethe Ansatz equations for this model and gave an interpretation of the spectra in terms of opers with an irregular singularity. In this paper, we consider a new geometric approach to the study of the spectra of the inhomogeneous Gaudin model in terms of a further enhancement of opers called twisted Miura-Plücker opers. This approach involves a certain system of nonlinear differential equations called the qq-system, which were previously studied in [20] in the context of the Bethe Ansatz. We show that there is a close relationship between solutions of the inhomogeneous Bethe Ansatz equations and polynomial solutions of the qq-system and use this fact to construct a bijection between the set of solutions of the inhomogeneous Bethe Ansatz equations and the set of nondegenerate twisted Miura-Plücker opers. We further prove that as long as certain combinatorial conditions are satisfied, nondegenerate twisted Miura-Plücker opers are in fact Miura opers.

Hubble diagrams in statistically homogeneous, anisotropic universes

We consider the form of Hubble diagrams that would be constructed by observers in universes that are homogeneous but anisotropic, when averaged over suitably large length-scales. This is achieved by ray-tracing in different directions on the sky in families of exact inhomogeneous cosmological solutions of Einstein's equations, in order to determine the redshifts and luminosity distances that observers in these space-times would infer for distant astrophysical objects. We compare the results of this procedure to the Hubble diagrams that would be obtained by direct use of the large-scale-averaged anisotropic cosmological models, and find that observables calculated in the averaged model closely agree with those obtained from ray-tracing in all cases where a statistical homogeneity scale exists. In contrast, we find that in cosmologies with spaces that contain no statistical homogeneity scale that Hubble diagrams inferred from the averaged cosmological model can differ considerably from those that observers in the space-time would actually construct. We hope that these results will be of use for understanding and interpreting recent observations that suggest that large-scale anisotropy may have developed in the late Universe.

Equivariant Hopf Bifurcation in a Class of Partial Functional Differential Equations on a Circular Domain

Circular domains frequently appear in mathematical modeling in the fields of ecology, biology and chemistry. In this paper, we investigate the equivariant Hopf bifurcation of partial functional differential equations with Neumann boundary condition on a two-dimensional disk. The properties of these bifurcations at equilibriums are analyzed rigorously by studying the equivariant normal forms. Two reaction–diffusion systems with discrete time delays are selected as numerical examples to verify the theoretical results, in which spatially inhomogeneous periodic solutions including standing waves and rotating waves, and spatially homogeneous periodic solutions are found near the bifurcation points.

Bifurcations of Spatially Inhomogeneous Solutions in a Modified Version of the Kuramoto–Sivashinsky Equation

Bifurcations of Spatially Inhomogeneous Solutions in a Modified Version of the Kuramoto–Sivashinsky Equation

Stable spatially inhomogeneous periodic solutions for a diffusive Leslie–Gower predator–prey model

Stable spatially inhomogeneous periodic solutions for a diffusive Leslie–Gower predator–prey model

Spatiotemporal Dynamics of a General Two-Species System with Taxis Term

In this paper, we investigate the spatiotemporal dynamics in a diffusive two-species system with taxis term and general functional response, which means the directional movement of one species upward or downward the other one. The stability of positive equilibrium and the existences of Turing bifurcation, Turing–Hopf bifurcation and Turing–Turing bifurcation are investigated. An algorithm for calculating the normal form of the Turing–Hopf bifurcation induced by the taxis term and another parameter is derived. Furthermore, we apply our theoretical results to a cooperative Lotka–Volterra system and a predator–prey system with prey-taxis. For the cooperative system, stable equilibrium becomes unstable by taxis-driven Turing instability, which is impossible for the cooperative system without taxis. For a predator–prey system with prey-taxis, the dynamical classification near the Turing–Hopf bifurcation point is clearly described. Near the Turing–Hopf point, there are spatially inhomogeneous steady-state solution, spatially homogeneous/nonhomogeneous periodic solution and pattern transitions from one spatiotemporal state to another one.

Spatiotemporal dynamics of a diffusive nutrient-phytoplankton model with delayed nutrient recycling

In this paper, we investigate the spatiotemporal dynamics of a diffusive nutrient-phytoplankton model with delayed nutrient recycling. We first study the stability of positive equilibrium and Turing instability induced by diffusion. We then investigate the effect of delay, and it turns out that the value of the rate of recycling k plays an important role in the Hopf bifurcation induced by delay. The delay will and will not induce Hopf bifurcation with low and high level of k, respectively. To reveal the spatiotemporal dynamics, Turing–Hopf bifurcation is carried out, and normal form is derived. Many spatiotemporal dynamics are found, including the coexistence of two stable spatially inhomogeneous periodic solutions or two stable spatially inhomogeneous steadystate solutions.

Spatiotemporal patterns induced by Turing-Hopf interaction and symmetry on a disk.

Turing bifurcation and Hopf bifurcation are two important kinds of transitions giving birth to inhomogeneous solutions, in spatial or temporal ways. On a disk, these two bifurcations may lead to equivariant Turing-Hopf bifurcations whose normal forms are given in three different cases in this paper. In addition, we analyzed the possible solutions for each normal form, which can guide us to find solutions with physical significance in real-world systems, and the breathing, standing wave-like, and rotating wave-like patterns are found in a delayed mussel-algae model.

Microstructural evolution and multi-mechanism strengthening model of nanocrystalline Al-Mg alloys

To unravel the determinants contributing to the remarkable strength observed in nanocrystalline binary alloys, we employed a high-pressure torsion method to fabricate Al-Mg alloys with varying Mg concentrations. Mechanical property analysis indicates that, with the increasing Mg content, both the hardness and yield strength exhibit an ascending trend. Notably, the nanocrystalline Al-8Mg alloy boasted an impressive hardness and yield strength of 295 HV and 889 MPa, respectively. Comprehensive characterizations of solute distribution and microstructural evolution were conducted using X-ray diffraction, electron back-scatter diffraction, transmission electron microscopy, and atom probe tomography. We clarified the distribution mechanism of Mg atoms at grain boundaries, developed a method for the precise assessment of inhomogeneous solute distribution, and proposed a multi-mechanism strengthening model, incorporating the inhomogeneous solute distribution strengthening mechanism. The calculated results aligned well with the experimental data.

Backpropagation in hyperbolic chaos via adjoint shadowing

To generalise the backpropagation method to both discrete-time and continuous-time hyperbolic chaos, we introduce the adjoint shadowing operator S acting on covector fields. We show that S can be equivalently defined as: S is the adjoint of the linear shadowing operator S; S is given by a ‘split then propagate’ expansion formula; S(ω) is the only bounded inhomogeneous adjoint solution of ω. By (a), S adjointly expresses the shadowing contribution, a significant part of the linear response, where the linear response is the derivative of the long-time statistics with respect to system parameters. By (b), S also expresses the other part of the linear response, the unstable contribution. By (c), S can be efficiently computed by the nonintrusive shadowing algorithm in Ni and Talnikar (2019 J. Comput. Phys. 395 690–709), which is similar to the conventional backpropagation algorithm. For continuous-time cases, we additionally show that the linear response admits a well-defined decomposition into shadowing and unstable contributions.

Hopf bifurcation induced by fear: A Leslie-Gower reaction-diffusion predator-prey model

<p>The aim of this paper was to explore the impact of fear on the dynamics of prey and predator species. Specifically, we investigated a reaction-diffusion predator-prey model in which the prey was subjected to Beddington-DeAngelis type and the predator was subjected to modified Leslie-Gower type. First, we analyzed the existence and stability of equilibria of the nonspatial model, and further investigated the global stability and Hopf bifurcation at the unique positive equilibrium point. For the spatial model, we studied the local and global stability of the unique constant positive steady state solution and captured the existence of Turing instability, which depended on the diffusion rate ratio between the two species. Then, we demonstrated the existence of Hopf bifurcations and discussed the direction and stability of spatially homogeneous and inhomogeneous periodic solutions. Finally, the impact of fear and spatial diffusion on the dynamics of populations were probed by numerical simulations. Results revealed that spatial diffusion and fear both broaden the dynamical properties of this model, facilitating the emergence of periodic solutions and the formation of biodiversity.</p>

Bifurcations in a General Delay Sel’kov–Schnakenberg Reaction–Diffusion System

The dynamics of a delay Sel’kov–Schnakenberg reaction–diffusion system are explored. The existence and the occurrence conditions of the Turing and the Hopf bifurcations of the system are found by taking the diffusion coefficient and the time delay as the bifurcation parameters. Based on that, the existence of codimension-2 bifurcations including Turing–Turing, Hopf–Hopf and Turing–Hopf bifurcations are given. Using the center manifold theory and the normal form method, the universal unfolding of the Turing–Hopf bifurcation at the positive constant steady-state is demonstrated. According to the universal unfolding, a Turing–Hopf bifurcation diagram is shown under a set of specific parameters. Furthermore, in different parameter regions, we find the existence of the spatially inhomogeneous steady-state, the spatially homogeneous and inhomogeneous periodic solutions. Discretization of time and space visualizes these spatio-temporal solutions. In particular, near the critical point of Hopf–Hopf bifurcation, the spatially homogeneous periodic and inhomogeneous quasi-periodic solutions are found numerically.