Periodic orbits of the ABC flow with A=B=C=1A=B=C=1

In this paper, we prove that the ODE system $$\begin{align*} \dot x &=\sin z+\cos y\\ \dot y &= \sin x+\cos z\\ \dot z &=\sin y + \cos x, \end{align*}$$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters $A=B=C=1$, has periodic orbits on $(2\pi\mathbb T)^3$ with rotation vectors parallel to $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $\mathbb R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.Comment: 9 page

Higher-order Szegő theorems with two singular points

We consider probability measures, dμ = w(θ)^(dθ)_(2π) + dμ_s, on the unit circle, ∂D, with Verblunsky coefficients, {αj}_(j=0)^∞. We prove for θ_1 ≠ θ_2 in [0,2π) that ∫[1-cos(θ-θ_1)][1-cos(θ-θ_2)]log w(θ)^(dθ)_(2π > -∞if and only if ∑_(j=0)^∞ │{(δ-e^(-iθ2))(δ-e^(-iθ1))α}_j^2 +|α_j|^4 - ∞ if and only if ∑_(j=0)^∞|α_(j+2) - 2α_(j+1) + α_j|^2 + |αj|^ 6 <∞

Ballistic Orbits and Front Speed Enhancement for ABC Flows

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the non-integrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular, the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.Comment: 39 pages, 26 figure

Pulsating Front Speed-up and Quenching of Reaction by Fast Advection

We consider reaction-diffusion equations with combustion-type non-linearities in two dimensions and study speed-up of their pulsating fronts by general periodic incompressible flows with a cellular structure. We show that the occurence of front speed-up in the sense $\lim_{A\to\infty} c_*(A)=\infty$, with $A$ the amplitude of the flow and $c_*(A)$ the (minimal) front speed, only depends on the geometry of the flow and not on the reaction function. In particular, front speed-up happens for KPP reactions if and only if it does for ignition reactions. We also show that the flows which achieve this speed-up are precisely those which, when scaled properly, are able to quench any ignition reaction.Comment: 16p

Multidimensional transition fronts for Fisher–KPP reactions

We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution 0 < u < 1 is known to satisfy lim t→−∞ sup|x|≤c|t| u(t,x) = 0 for each c < 2√f′(0), and we consider here those satisfying lim t→−∞ sup|x|≤c|t| u(t,x) = 0 for some c > 2√f′(0). When f is C_2 and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions