Propagation Of Singularities Research Articles

Simulation of self-healing of polarization singularities

The generation and propagation of polarization singularities have been researched for years. In this article, we explored self-healing of polarization singularity generated from Bessel beams when encountering obstacles during propagation. The topological structure can be reconstructed due to the self-healing of Bessel beams, and we found that the reconstruction of the topological structure is much faster than the healing of the broken amplitudes.

Generalized characteristics and Lax–Oleinik operators: global theory

For autonomous Tonelli systems on $$\mathbb {R}^n$$ , we develop an intrinsic proof of the existence of generalized characteristics using sup-convolutions. This approach, together with convexity estimates for the fundamental solution, leads to new results such as the global propagation of singularities along generalized characteristics.

Propagation of Correlation Singularities of a Partially Coherent Laguerre–Gaussian Electromagnetic Beam in a Uniaxial Crystal

The correlation singularities of a partially coherent Laguerre–Gaussian (LG) electromagnetic beam in a uniaxial crystal is shown to possess interesting evolution behavior. The analytical formula for the cross-spectral density matrix of a partially coherent LG electromagnetic beam propagating in a uniaxial crystal orthogonal to the optical axis is derived. It is found that the correlation singularities of a partially coherent LG electromagnetic beam in a uniaxial crystal is closely determined by the spatial coherence and mode orders of the beam, and the refractive indexes of the crystal. For a partially coherent LG electromagnetic beam, the correlation singularities can propagate further in a uniaxial crystal than that in isotropic medium. For a fully coherent LG electromagnetic beam, in isotropic medium the central correlation singularities and circular edge dislocations persist upon propagation; in a uniaxial crystal the circular edge dislocations become imperfect, and the central correlation singularities with topological charges $l \ne 1$ split up into correlation singularities with topological charges equaling 1.

Propagation of Gabor singularities for Schrödinger equations with quadratic Hamiltonians

AbstractWe study propagation of the Gabor wave front set for a Schrödinger equation with a Hamiltonian that is the Weyl quantization of a quadratic form with nonnegative real part. We point out that the singular space associated with the quadratic form plays a crucial role for the understanding of this propagation. We show that the Gabor singularities of the solution to the equation for positive times are always contained in the singular space, and that they propagate in this set along the flow of the Hamilton vector field associated with the imaginary part of the quadratic form. As an application we obtain for the heat equation a sufficient condition on the Gabor wave front set of the initial datum tempered distribution that implies regularization to Schwartz regularity for positive times.

Partial exact controllability for wave equations

We analyze the partial exact controllability problem for wave equations. The goal is to drive part of the solution to a destination at a given time. The main results are proved by employing the classical theory of propagation of singularities of wave equations and the compact uniqueness argument.

Reducing Streaking Artifacts in Quantitative Susceptibility Mapping

It is well known that reconstruction algorithms in quantitative susceptibility mapping often contain streaking artifacts. These are nondesirable objects that contaminate the image, and the possibility of removing or at least reducing them has a great practical interest. In [J. K. Choi, H. S. Park, S. Wang, Y. Wang, and J. K. Seo, SIAM J. Imaging Sci., 7 (2014), pp. 1669--1689], the cause of the artifacts is identified as propagation of singularities for a wave-type operator. In this work, we analyze such singularities using microlocal techniques and propose some strategies to reduce the artifacts.

Inhomogeneous microlocal propagation of singularities in Fourier Lebesgue spaces

In the paper some results of microlocal continuity for pseudodifferential operators whose symbols belong to weighted Fourier Lebesgue spaces are given. Inhomogeneous local and microlocal propagation of singularities of Fourier Lebesgue type are then studied, with applications to some classes of semilinear equations.

Diffractive Propagation on Conic Manifolds

In this survey, we review some applications and extensions of the author’s results with Richard Melrose on propagation of singularities for solutions to the wave equation on manifolds with conical singularities. These results mainly concern: the local decay of energy on noncompact manifolds with diffractive trapped orbits (joint work with Dean Baskin); singularities of the wave trace created by diffractive closed geodesics (joint work with G. Austin Ford); and the distribution of scattering resonances associated to such closed geodesics (joint work with Luc Hillairet).

Propagation of strong singularities in semilinear parabolic equations with degenerate absorption

Propagation of strong singularities in semilinear parabolic equations with degenerate absorption

Microlocal analysis of fractional wave equations

We determine the wave front sets of solutions to two special cases of the Cauchy problem for the space‐time fractional Zener wave equation, one being fractional in space, the other being fractional in time. For the case of the space fractional wave equation, we show that no spatial propagation of singularities occurs. For the time fractional Zener wave equation, we show an analogue of non‐characteristic regularity.

100 years of Weyl’s law

We discuss the asymptotics of the eigenvalue counting function for partial differential operators and related expressions paying the most attention to the sharp asymptotics. We consider Weyl asymptotics, asymptotics with Weyl principal parts and correction terms and asymptotics with non-Weyl principal parts. Semiclassical microlocal analysis, propagation of singularities and related dynamics play crucial role. We start from the general theory, then consider Schr\"odinger and Dirac operators with the strong magnetic field and, finally, applications to the asymptotics of the ground state energy of heavy atoms and molecules with or without a magnetic field.

Superposition of nonparaxial vectorial complex-source spherically focused beams: Axial Poynting singularity and reverse propagation

In this work, counterintuitive effects such as the generation of an axial (i.e., long the direction of wave motion) zero-energy flux density (i.e., axial Poynting singularity) and reverse (i.e., negative) propagation of nonparaxial quasi-Gaussian electromagnetic (EM) beams are examined. Generalized analytical expressions for the EM field's components of a coherent superposition of two high-order quasi-Gaussian vortex beams of opposite handedness and different amplitudes are derived based on the complex-source-point method, stemming from Maxwell's vector equations and the Lorenz gauge condition. The general solutions exhibiting unusual effects satisfy the Helmholtz and Maxwell's equations. The EM beam components are characterized by nonzero integer degree and order $(n,m)$, respectively, an arbitrary waist ${w}_{0}$, a diffraction convergence length known as the Rayleigh range ${z}_{R}$, and a weighting (real) factor $\phantom{\rule{4pt}{0ex}}0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}1$ that describes the transition of the beam from a purely vortex $(\ensuremath{\alpha}=0)$ to a nonvortex $(\ensuremath{\alpha}=1)$ type. An attractive feature for this superposition is the description of strongly focused (or strongly divergent) wave fields. Computations of the EM power density as well as the linear and angular momentum density fluxes illustrate the analysis with particular emphasis on the polarization states of the vector potentials forming the beams and the weight of the coherent beam superposition causing the transition from the vortex to the nonvortex type. Should some conditions determined by the polarization state of the vector potentials and the beam parameters be met, an axial zero-energy flux density is predicted in addition to a negative retrograde propagation effect. Moreover, rotation reversal of the angular momentum flux density with respect to the beam handedness is anticipated, suggesting the possible generation of negative (left-handed) torques. The results are particularly useful in applications involving the design of strongly focused optical laser tweezers, tractor beams, optical spanners, arbitrary scattering, radiation force, angular momentum, and torque in particle manipulation, and other related topics.

Global propagation of singularities for solutions of Hamilton–Jacobi equations

We show that the singularities of the viscosity solutions for a class of Hamilton–Jacobi equations propagate along the generalized characteristics for all the times.

Some counterexamples to Sobolev regularity for degenerate Monge–Ampère equations

We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial \Omega} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.

Wigner measures and observability for the Schrödinger equation on the disk

We analyse the structure of semiclassical and microlocal Wigner measures for solutions to the linear Schrödinger equation on the disk, with Dirichlet boundary conditions. Our approach links the propagation of singularities beyond geometric optics with the completely integrable nature of the billiard in the disk. We prove a “structure theorem”, expressing the restriction of the Wigner measures on each invariant torus in terms of second-microlocal measures. They are obtained by performing a finer localization in phase space around each of these tori, at the limit of the uncertainty principle, and are shown to propagate according to Heisenberg equations on the circle. Our construction yields as corollaries (a) that the disintegration of the Wigner measures is absolutely continuous in the angular variable, which is an expression of the dispersive properties of the equation; (b) an observability inequality, saying that the $$L^2$$ -norm of a solution on any open subset intersecting the boundary (resp. the $$L^2$$ -norm of the Neumann trace on any nonempty open set of the boundary) controls its full $$L^2$$ -norm (resp. $$H^1$$ -norm). These results show in particular that the energy of solutions cannot concentrate on periodic trajectories of the billiard flow other than the boundary.

Propagation of Exponential Phase Space Singularities for Schrödinger Equations with Quadratic Hamiltonians

We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrödinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand–Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand–Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.

A Quantitative Vainberg Method for Black Box Scattering

We give a quantitative version of Vainberg's method relating pole free regions to propagation of singularities for black box scatterers. In particular, we show that there is a logarithmic resonance free region near the real axis of size $\tau$ with polynomial bounds on the resolvent if and only if the wave propagator gains derivatives at rate $\tau$. Next we show that if there exist singularities in the wave trace at times tending to infinity which smooth at rate $\tau$, then there are resonances in logarithmic strips whose width is given by $\tau$. As our main application of these results, we give sharp bounds on the size of resonance free regions in scattering on geometrically nontrapping manifolds with conic points. Moreover, these bounds are generically optimal on exteriors of nontrapping polygonal domains.

M‐Microlocal elliptic pseudodifferential operators acting on

In the first part of the paper we study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fréchet space . In the second one non homogeneous microlocal properties are introduced and propagation of Sobolev singularities for solutions to (pseudo)differential equations is given. For both the arguments actual examples are provided.

Lasry-Lions, Lax-Oleinik and generalized characteristics

In the recent works \cite{Cannarsa-Chen-Cheng} and \cite{Cannarsa-Cheng3}, an intrinsic approach of the propagation of singularities along the generalized characteristics was obtained, even in global case, by a procedure of sup-convolution with the kernel the fundamental solutions of the associated Hamilton-Jacobi equations. In the present paper, we exploit the relations among Lasry-Lions regularization, Lax-Oleinik operators (or inf/sup-convolution) and generalized characteristics, which are discussed in the context of the variational setting of Tonelli Hamiltonian dynamics, such as Mather theory and weak KAM theory.

A new proof of indefinite propagation of singularities for a Hamilton–Jacobi equation

We study propagation of singularities for the Hamilton–Jacobi equation $$S_t+H(\nabla S) = 0,\quad (t,x) \in (0,T) \times \mathbb{R}^n,$$ where \({H(p)=\frac{1}{2} \langle p,Ap\rangle}\) is a positive definite quadratic form. Each viscosity solution \({S}\) is semiconcave, and it is known that its singularities move along generalized characteristics. We give a new proof of the recent result by Cannarsa et al. (Discrete Contin Dyn Syst 35:4225–4239, 2015), namely that the singularities propagate along generalized characteristics indefinitely forward in time.