arXiv subjects

Jonáš Volek

Publications and source records attributed to Jonáš Volek.

3 recordsLinked to original sources

arXiv2026

Characterization of spatial topological chaos for bistable lattice equation

The propagation failure phenomenon in bistable lattice equations connects spatial discreteness and weak diffusion to the large number of stationary patterns and the inability of waves to move. We show that there naturally exist bistable lattice equations that do not admit propagation failure for intermediate values of diffusion parameters. The collapse of propagation failure is connected with the existence of a continuum of pinned waves and a continuum of bounded nonmonotone stationary solutions. It also enables us to find explicit forms of traveling waves of families of bistable lattice equations.

math.DS↗

arXiv2024

Perfect stationary solutions of reaction-diffusion equations on lattices and regular graphs

Reaction-diffusion equations on infinite graphs can have an infinite number of stationary solutions. These solutions are generally described as roots of a countable system of algebraic equations. As a generalization of periodic stationary solutions we present perfect stationary solutions, a special class of solutions with finite range in which the neighborhood values are determined precisely by the value of the central vertex. The focus on the solutions which attain a finite number of values enables us to rewrite the countable algebraic system to a finite one. In this work, we define the notion of perfect stationary solutions and show its elementary properties. We further present results from the theory of perfect colorings in order to prove the existence of the solutions in the square, triangular and hexagonal grids; as a byproduct, the existence of uncountable number of two-valued stationary solutions on these grids is shown. These two-valued solutions can form highly aperiodic and highly irregular patterns. Finally, an application to a bistable reaction-diffusion equation on a square grid is presented.

math.DS↗

arXiv2012

Transport Equation on Semidiscrete Domains and Poisson-Bernoulli Processes

In this paper we consider a scalar transport equation with constant coefficients on domains with discrete space and continuous, discrete or general time. We show that on all these underlying domains solutions of the transport equation can conserve sign and integrals both in time and space. Detailed analysis reveals that, under some initial conditions, the solutions correspond to counting stochastic processes and related probability distributions. Consequently, the transport equation could generate various modifications of these processes and distributions and provide some insights into corresponding convergence questions. Possible applications are suggested and discussed.

math.AP↗