arXiv subjects

Alexei Cheviakov

Publications and source records attributed to Alexei Cheviakov.

9 recordsLinked to original sources

arXiv2026

Nonlinear Incompressible Shear Wave Models in Hyperelasticity and Viscoelasticity Frameworks, with Applications to Love Waves

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For a cubic Yeoh model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives $u_{xxt}$ of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analyzed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition $c_1 < \abs{v} < c_2$, while tending to the larger material wave speed $c_1$ or $c_2$ for large times.

nlin.SI↗

arXiv2026

Invariant Reduction for Partial Differential Equations. IV: Symmetries that Rescale Geometric Structures

For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic structures, variational principles, and Poisson brackets, are inherited by the systems governing symmetry-invariant solutions. We extend this mechanism to geometric structures that are not invariant but are $\textit{rescaled}$ by a symmetry. Specifically, if $X$ is the symmetry used for reduction, $X_s$ is a symmetry satisfying $[X_s,X]=aX$, and the Lie derivative $\mathcal{L}_{X_s}$ acts on an $X$-invariant element of the Vinogradov $\mathcal{C}$-spectral sequence as multiplication by $b$, then the restricted symmetry $X_s|_{\mathcal{E}_X}$ acts on the corresponding reduction as multiplication by $a+b$. This shift rule gives rise to two phenomena: the $\textit{emergence of invariance}$, where reductions acquire an invariance that was not present at the level of the original structure, and the $\textit{loss of invariance}$, where reductions of invariant structures are no longer invariant. As an application, we describe a class of exact solutions to systems possessing sufficiently many symmetries and conservation laws subject to certain compatibility conditions. These solutions are invariant under pairs of symmetries and are completely determined by explicitly constructed functions that are constant on them; the description is geometric and does not require any integrability-related structures such as Lax pairs. The framework is illustrated by two examples: the Lin--Reissner--Tsien equation of potential nonstationary transonic gas flows, for which closed-form exact solutions are obtained and validated numerically, and the potential Boussinesq system, for which the inherited Poisson bracket is employed to describe solutions determined by algebraic equations.

nlin.SI↗

arXiv2025

Analytical solutions for the Extracellular-Membrane-Intracellular model

The Extracellular-Membrane-Intracellular (EMI) model is a novel mathematical framework for cardiac electrophysiology simulations. The EMI model provides a more detailed description of the heart's electrical activity compared to traditional monodomain and bidomain models, potentially making it better-suited for understanding the electrical dynamics of the heart under pathological conditions. In this paper, we derive and verify several analytical solutions for the EMI model. Specifically, we obtain a family of solutions for a single two-dimensional cell in polar coordinates and for a pair of coupled three-dimensional cells in spherical coordinates. We also introduce a manufactured solution for N three-dimensional cells in Cartesian coordinates. To verify the analytical solutions, we conduct numerical experiments using the mortar finite element method combined with operator splitting. The results demonstrate that the analytical solutions are effective for verifying the accuracy of numerical simulations of the EMI model.

math.NA↗

arXiv2025

Invariant Reduction for Partial Differential Equations. II: The General Framework

For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $\varphi$, invariant solutions satisfy the reduced system $F = \varphi = 0$. We propose a framework that allows, for every $X$-invariant conservation law, presymplectic structure, variational principle, or another geometric structure of the given PDE system $F = 0$, to systematically calculate its corresponding reduced form that describes the corresponding structure for the reduced system $F = \varphi = 0$. In particular, we show in what way Noether's theorem holding for the given PDE system is inherited by the reduced PDE system. We consider several detailed examples, including cases of point and higher symmetry invariance. The proposed framework is directly applicable to a wide range of PDE models, including complex PDE systems of contemporary interest arising across disciplines, where symmetry reduction is essential for analysis and simulation, as well as to integrable, Lagrangian, and gauge systems.

nlin.SI↗

arXiv2024

Invariant Reduction for Partial Differential Equations. I: Conservation Laws and Systems with Two Independent Variables

For a system of partial differential equations that has an extended Kovalevskaya form, a reduction procedure is presented that allows one to use a local (point, contact, or higher) symmetry of a system and a symmetry-invariant conservation law to algorithmically calculate constants of motion holding for symmetry-invariant solutions. Several examples including cases of point and higher symmetry invariance are presented and discussed. An implementation of the algorithm in Maple is provided.

nlin.SI↗

arXiv2024

Spatiotemporal Behaviour of SIR Models with Cross-Diffusion and Vital Dynamics

Contemporary epidemiological models often involve spatial variation, providing an avenue to investigate the averaged dynamics of individual movements. In this work, we extend a recent model by Vaziry, Kolokolnikov, and Kevrekidis [Royal Society Open Science 9 (10), 2022] that included, in both infected and susceptible population dynamics equations, a cross-diffusion term with the second spatial derivative of the infected population density. Diffusion terms of this type occur, for example, in the Keller-Siegel chemotaxis model. The presented model corresponds to local orderly commute of susceptible and infected individuals, and is shown to arise in two dimensions as a limit of a discrete process. The present contribution identifies and studies specific features of the new model's dynamics, including various types of infection waves and buffer zones protected from the infection. The model with vital dynamics additionally exhibits complex spatiotemporal behaviour that involves the generation of quasiperiodic infection waves and emergence of transient strongly heterogeneous patterns.

q-bio.PE↗

arXiv2021

Narrow Escape Brownian Dynamics Modeling in the Three-Dimensional Unit Sphere

The narrow escape problem is a first-passage problem concerned with randomly moving particles in a physical domain, being trapped by absorbing surface traps (windows), such that the measure of traps is small compared to the domain size. The expected value of time required for a particle to escape is defined as mean first passage time (MFPT), which satisfies the Poisson partial differential equation subject to a mixed Dirichlet-Neumann boundary condition. The primary objective of this work is a direct numerical simulation of multiple particles undergoing Brownian motion in a three-dimensional sphere with boundary traps, compute MFPT values by averaging Brownian escape times, and compare the results with asymptotic results obtained by solving the Poisson PDE problem. A comprehensive study of results obtained from the simulations shows that the difference between Brownian and asymptotic results for the escape times mostly not exceed $1\%$ accuracy. This comparison in some sense validates the narrow escape PDE problem itself as an approximation (averaging) of the multiple physical Brownian motion runs. This work also predicted that how many single-particle simulations are required to match the predicted asymptotic averaged MFPT values. The next objective of this work is to study dynamics of Brownian particles near the boundary by estimating the average percentage of time spent by Brownian particle near the domain boundary for both the anisotropic and isotropic diffusion. It is shown that the Brownian particles spend more in the boundary layer than predicted by the boundary layer relative volume, with the effect being more pronounced in a narrow layer near the spherical wall. It is also shown that taking into account anisotropic diffusion yields larger times a particle spends near the boundary, and smaller escape times than those predicted by the isotropic diffusion model.

math-ph↗

arXiv2021

Global Optimization of the Mean First Passage Time for Narrow Capture Problems in Elliptic Domains

Narrow escape and narrow capture problems which describe the average times required to stop the motion of a randomly travelling particle within a domain have applications in various areas of science. While for general domains, it is known how the escape time decreases with the increase of the trap sizes, for some specific 2D and 3D domains, higher-order asymptotic formulas have been established, providing the dependence of the escape time on the sizes and locations of the traps. Such results allow the use of global optimization to seek trap arrangements that minimize average escape times. In a recent paper \cite{iyaniwura2021optimization}, an explicit size- and trap location-dependent expansion of the average mean first passage time (MFPT) in a 2D elliptic domain was derived. The goal of this work is to systematically seek global minima of MFPT for $1\leq N\leq 50$ traps in elliptic domains using global optimization techniques, and compare the corresponding putative optimal trap arrangements for different values of the domain eccentricity. Further, an asymptotic formula the for the average MFPT in elliptic domains with $N$ circular traps of arbitrary sizes is derived, and sample optimal configurations involving non-equal traps are computed.

cond-mat.stat-mech↗

arXiv2019

Radial waves in fiber-reinforced axially symmetric hyperelastic media

Complex elastic media such as biological membranes, in particular, blood vessels, may be described as fiber-reinforced solids in the framework of nonlinear hyperelasticity. Finite axially symmetric anti-plane shear displacements in such solids are considered. A general nonlinear wave equation governing such motions is derived. It is shown that in the case of Mooney-Rivlin materials with standard quadratic fiber energy term, the displacements are governed by a linear cylindrical wave equation. Extensions of the model onto the case when fibers have a radial projection, as well as onto a viscoelastic case taking into account dissipative effects, are considered; wave equations governing shear displacements in those cases are derived and analyzed.

physics.class-ph↗