We carried out a computational study of propagation speeds ofreaction-diffusion-advection fronts in three dimensional (3D) cellular andArnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP)nonlinearity. The variational principle of front speeds reduces the problem to a principaleigenvalue calculation. An adaptive streamline diffusion finite element method is used inthe advection dominated regime. Numerical results showed that the front speeds areenhanced in cellular flows according to sublinear power lawO(δp),p ≈ 0.13, δ the flow intensity. In ABC flows however,the enhancement is O(δ) which can be attributed to thepresence of principal vortex tubes in the streamlines. Poincaré sections are used tovisualize and quantify the chaotic fractions of ABC flows in the phase space. The effectof chaotic streamlines of ABC flows on front speeds is studied by varying the threeparameters (a,b,c) of the ABC flows. Speed enhancement alongx direction is reduced as b (the parameter controlingthe flow variation along x) increases at fixed(a,c) > 0, more rapidly as the corresponding ABC streamlines becomemore chaotic.

Building on earlier work, we have given in [29] aproof of existence of Abrikosov vortex lattices in the Ginzburg-Landau model ofsuperconductivity and shown that the triangular lattice gives the lowest energy perlattice cell. After [29] was published, we realizedthat it proves a stronger result than was stated there. This result is recorded in thepresent paper. The proofs remain the same as in [29], apart from some streamlining.

Assuming the negative part of the potential is uniformly locallyL1, we prove a pointwiseLp estimate on derivatives ofeigenfunctions of one-dimensional Schrödinger operators. In particular, if aneigenfunction is in Lp, then so is itsderivative, for 1 ≤ p ≤ ∞.

We perform a computationl study of front speeds of G-equation models in time dependentcellular flows. The G-equations arise in premixed turbulent combustion, and areHamilton-Jacobi type level set partial differential equations (PDEs). The curvature-strainG-equations are also non-convex with degenerate diffusion. The computation is based onmonotone finite difference discretization and weighted essentially nonoscillatory (WENO)methods. We found that the large time front speeds lock into the frequency of timeperiodic cellular flows in curvature-strain G-equations similar to what occurs in thebasic inviscid G-equation. However, such frequency locking phenomenon disappears inviscous G-equation, and in the inviscid G-equation if time periodic oscillation of thecellular flow is replaced by time stochastic oscillation.

The model we consider treats the cell as a viscoelastic medium filling one of two kindsof thin domains (“shapes” of cells): the thin slab being a caricature of a tissue and thethin circular cylinder mimicking a long cell. This enables us to simplify the system ofmechano-chemical equations. We construct abundant classes of explicit, but approximate,formulae for heteroclinic solutions to these equations.

The slow dynamics and linearized stability of a two-spike quasi-equilibrium solution to ageneral class of reaction-diffusion (RD) system with and without sub-diffusion isanalyzed. For both the case of regular and sub-diffusion, the method of matched asymptoticexpansions is used to derive an ODE characterizing the spike locations in the absence ofany 𝒪(1) time-scale instabilities of the two-spike quasi-equilibrium profile. These fastinstabilities result from unstable eigenvalues of a certain nonlocal eigenvalue problem(NLEP) that is derived by linearizing the RD system around the two-spike quasi-equilibriumsolution. For a particular sub-class of the reaction kinetics, it is shown that thediscrete spectrum of this NLEP is determined by the roots of some simple transcendentalequations. From a rigorous analysis of these transcendental equations, explicit sufficientconditions are given to predict the occurrence of either Hopf bifurcations or competitioninstabilities of the two-spike quasi-equilibrium solution. The theory is illustrated forseveral specific choices of the reaction kinetics.

We introduce new classes of modulation spaces over phase space. By means of theKohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operatorsthat bound their operator norms on Lp–spaces,Sobolev spaces, and modulation spaces.

The heat equation is considered in the complex system consisting of many small bodies(particles) embedded in a given material. On the surfaces of the small bodies aNewton-type boundary condition is imposed. An equation for the limiting field is derivedwhen the characteristic size a of the small bodies tends to zero, theirtotal number \hbox{$\mathcal{N}(a)$}𝒩(a) tends to infinity at a suitable rate, and the distanced = d(a) between neighboring smallbodies tends to zero a <  < d. No periodicity isassumed about the distribution of the small bodies.

Inverse scattering problem is discussed for the Maxwell’s equations. A reduction of theMaxwell’s system to a new Fredholm second-kind integral equation with a scalarweakly singular kernel is given for electromagnetic (EM) wave scattering. Thisequation allows one to derive a formula for the scattering amplitude in which only ascalar function is present. If this function is small (an assumption that validates aBorn-type approximation), then formulas for the solution to the inverse problem areobtained from the scattering data: the complex permittivityϵ′(x) in a bounded regionD ⊂ R3 is found from the scattering amplitudeA(β,α,k) known for a fixed k = ω √ϵ0μ0 >0 and allβ,α ∈ S2, whereS2 is the unit sphere in R3,ϵ0 and μ0 are constantpermittivity and magnetic permeability in the exterior regionD′ = R3\D. The novel pointsin this paper include: i) A reduction of the inverse problem for vectorEM waves to a vector integral equation with scalar kernelwithout any symmetry assumptions on the scatterer, ii) A derivation of thescalar integral equation of the first kind for solving the inversescattering problem, and iii) Presenting formulas for solving this scalar integralequation. The problem of solving this integral equation is an ill-posed one. A method fora stable solution of this problem is given.

Let L be a linear, closed, densely defined in a Hilbert space operator,not necessarily selfadjoint. Consider the corresponding wave equations

\begin{eqnarray} &(1) \quad \ddot{w}+ Lw=0, \quadw(0)=0,\quad \dot{w}(0)=f, \quad \dot{w}=\frac{dw}{dt}, \quad f \in H. \\ &(2)\quad \ddot{u}+Lu=f e^{-ikt}, \quad u(0)=0, \quad \dot{u}(0)=0,\end{eqnarray}$\begin{array}{ccc}& \mathrm{\left(}\mathrm{1}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{¨}\\ \mathit{w}\end{array}\mathrm{+}\mathit{Lw}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{w}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{ẇ}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathit{f,}\hspace{1em}\mathit{ẇ}\mathrm{=}\frac{\mathit{dw}}{\mathit{dt}}\mathit{,}\hspace{1em}\mathit{f}\mathrm{\in }\mathit{H}\mathit{.}& \\ & \mathrm{\left(}\mathrm{2}\mathrm{\right)}\hspace{1em}\begin{array}{}\mathrm{¨}\\ \mathit{u}\end{array}\mathrm{+}\mathit{Lu}\mathrm{=}\mathit{f}{\mathit{e}}^{\mathrm{-}\mathit{ikt}}\mathit{,}\hspace{1em}\mathit{u}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}\hspace{1em}\mathit{u̇}\mathrm{\left(}\mathrm{0}\mathrm{\right)}\mathrm{=}\mathrm{0}\mathit{,}& \end{array}$

where k > 0 is a constant. Necessary and sufficient conditions aregiven for the operator L not to have eigenvalues in the half-planeRez < 0 and not to have a positive eigenvalue at a given point kd2 > 0. These conditions are given in terms of the large-timebehavior of the solutions to problem (1) for generic f.

Sufficient conditions are given for the validity of a version of the limiting amplitudeprinciple for the operator L.

A relation between the limiting amplitude principle and the limiting absorption principleis established.

Stockwell transforms as hybrids of Gabor transforms and wavelet transforms have beenstudied extensively. We introduce in this paper multi-dimensional Stockwell transformsthat include multi-dimensional Gabor transforms as special cases. Continuous inversionformulas for multi-dimensional Stockwell transforms are proved.

In this paper, an open problem in the multidimensional complex analysis is presented thatarises in the harmonic analysis related to the investigation of the regularity propertiesof Fourier integral operators and in the regularity theory for hyperbolic partialdifferential equations. The problem is discussed in a self-contained elementary way andsome results towards its resolution are presented. A conjecture concerning the structureof appearing affine fibrations is formulated.

We establish sharp semiclassical upper bounds for the moments of some negative powers forthe eigenvalues of the Dirichlet Laplacian. When a constant magnetic field is incorporatedin the problem, we obtain sharp lower bounds for the moments of positive powers notexceeding one for such eigenvalues. When considering a Schrödinger operator with therelativistic kinetic energy and a smooth, nonnegative, unbounded potential, we prove thesharp Lieb-Thirring estimate for the moments of some negative powers of itseigenvalues.