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In this Series, we study the weakly nonlinear dynamics of chemically active particles near the threshold for spontaneous motion. In this part, we focus on steady solutions and develop an `adjoint method' for deriving the nonlinear amplitude…
We expand the off-resonant scattering theory for particle diffusion in magnetized current filaments that can be typically compared to astrophysical jets, including active galactic nucleus jets. In a high plasma beta region where the…
Shocks in astrophysical fluids can generate suprathermal particles by first order (or diffusive) Fermi acceleration. In the test particle regime there is a simple relation between the spectrum of the accelerated particles and the jump…
This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A…
The dynamical reaction of the particles accelerated at a shock front by the first order Fermi process can be determined within kinetic models that account for both the hydrodynamics of the shocked fluid and the transport of the accelerated…
We suggest a physical mechanism whereby the acceleration time of cosmic rays by shock waves can be significantly reduced. This creates the possibility of particle acceleration beyond the knee energy at ~10^15eV. The acceleration results…
Travelling wave solutions of reaction-diffusion equations are widely used to model the spatial spread of populations and other phenomena in biology and physics. In this article, we reinterpret the classical variational principle approach…
Systems consisting of a single ordinary differential equation coupled with one reaction-diffusion equation in a bounded domain and with the Neumann boundary conditions are studied in the case of particular nonlinearities from the…
Diffusive shock acceleration (DSA) is now widely accepted as the model to explain the production of cosmic rays (CRs) in a wide range of astrophysical environments. Despite initial successes of the theory in explaining the energetics and…
Context. The diffusive shock acceleration mechanism has been widely accepted as the acceleration mechanism for galactic cosmic rays. While self-consistent hybrid simulations have shown how power-law spectra are produced, detailed…
Turbulent magnetic fields are to some extent a universal feature in astrophysical phenomena. Charged particles that encounter these turbulence get on average accelerated according to the so-called second-order Fermi process. However, in…
This paper considers the solution structure of non-trivial, non-constant stationary states of 1D spatial parabolic equations with nonlinear self-diffusion and logistic growth terms. A two-dimensional ordinary differential equation…
In this article we address the problem of the nonlinear interaction of subdiffusive particles. We introduce the random walk model in which statistical characteristics of a random walker such as escape rate and jump distribution depend on…
In this paper we present the first formulation of the theory of non-linear particle acceleration in collisionless shocks in the presence of neutral hydrogen in the acceleration region. The dynamical reaction of the accelerated particles,…
In turbulent magnetized plasmas, charged particles can be accelerated to high energies through their interactions with the turbulent motions. As they do so, they draw energy from the turbulence, possibly up to the point where they start…
We study multiplicity of the supercritical traveling front solutions for scalar reaction-diffusion equations in infinite cylinders which invade a linearly unstable equilibrium. These equations are known to possess traveling wave solutions…
The main subject of this paper is a computer assisted stability proof for a stationary solution of reaction diffusion equations in one dimensional space. We use Nakao's numerical verification method to enclose a stationary solution of…
We study the spectral stability of travelling and stationary front and pulse solutions in a class of degenerate reaction-diffusion systems. We characterise the essential spectrum of the linearised operator in full generality and identify…
In this work, we shall consider the existence and uniqueness of stationary solutions to stochastic partial functional differential equations with additive noise in which a neutral type of delay is explicitly presented. We are especially…
We construct solutions of nonlinear reaction-diffusion equations with nonlinear boundary conditions in spaces where the problem is supercritical and show the nonlinear balance required between the nonlinear terms in order to obtain a…