arXiv subjects

Alex Mahalov

Publications and source records attributed to Alex Mahalov.

10 recordsLinked to original sources

arXiv2026

Dynamic Doppler Effects on Dirac Spinor Fields

We derive the dynamic Doppler effects of noninertial observer motion on Dirac spinor fields, characterizing how proper acceleration, proper jerk, observer path curvature, torsion, and hyper-torsion modify the inertial quantum description of free particles. We quantify the nonlinear behavior of the amplitude and phase of the Dirac spinor wave field and isolate the spin-induced signatures that have no counterpart in the scalar Klein-Gordon field. For worldlines with constant proper jerk the amplitude scales super-exponentially as $\exp(j_0\tau^2/4)$. The complex Frenet-Serret curvature invariants generate both an exponential amplitude factor and a spin-induced phase factor. Spinor structure therefore produces an observable dynamic Doppler signature distinct from any scalar prediction.

math-ph↗

arXiv2026

The Dynamic Doppler Spectrum Induced by Nonlinear Sensor Motion: Relativistic Kinematics and 4D Frenet-Serret Spacetime Geometry

Fundamental to the analysis of nonlinear relativistic motion is the precise characterization of the induced dynamic Doppler effects. In this work, we analyze the electromagnetic signals observed by non-inertial receivers using two frameworks to describe the relativistic motion. We first consider observer paths described by higher-order kinematic 4 vectors: relativistic acceleration and jolt. The dynamic Doppler effects of relativistic acceleration and jolt are exponential spectral broadening and exponential amplitude growth or decay. We derive compact expressions for the spectrum transformation resulting from relativistic acceleration and jolt. The jolt induces nonlinear skewed chirps in observed signals. Next we consider observer paths described by the 4D Frenet-Serret frame and the curvature and torsion of the observer path. We obtain descriptions of the amplitude and phase fluctuations of the signal in terms of the geometric parameters of curvature and torsion. Concise, interpretable descriptions of non-inertial dynamic Doppler effects provide a useful diagnostic and predictive tool for engineering applications including radar, sensing, and communications systems.

math-ph↗

arXiv2024

The Reconstruction of the Space-Dependent Thermal Conductivity from Sparse Temperature Measurements

We present a novel method for reconstructing the thermal conductivity coefficient in 1D and 2D heat equations using moving sensors that dynamically traverse the domain to record sparse and noisy temperature measurements. We significantly reduce the computational cost associated with forward PDE evaluations by employing automatic differentiation, enabling a more efficient and scalable reconstruction process. This allows the inverse problem to be solved with fewer sensors and observations. Specifically, we demonstrate the successful reconstruction of thermal conductivity on the 1D circle and 2D torus, using one and four moving sensors, respectively, with their positions recorded over time. Our method incorporates sampling algorithms to compute confidence intervals for the reconstructed conductivity, improving robustness against measurement noise. Extensive numerical simulations of heat dynamics validate the efficacy of our approach, confirming both the accuracy and stability of the reconstructed thermal conductivity. Additionally, the method is thoroughly tested using large datasets from machine learning, allowing us to evaluate its performance across various scenarios and ensure its reliability. This approach provides a cost-effective and flexible solution for conductivity reconstruction from sparse measurements, making it a robust tool for solving inverse problems in complex domains.

math.NA↗

arXiv2018

Coupling of paraxial and white-noise approximations of the Helmholtz equation in randomly layered media

We study the simultaneous paraxial and white-noise limit of the Helmholtz equation in randomly layered media where the refractive index fluctuations are in the direction of propagation. We consider the regime in which the wavelength is of the same order as the correlation length of the random fluctuations of the refractive index. We show that this simultaneous limit can be taken in this regime by introducing into the equation an arbitrarily small regularization parameter. The corresponding paraxial white-noise approximation that we derive is different from that of the previously studied high-frequency regime. Since the correlation length of the refractive index fluctuations due to atmospheric turbulence varies substantially, our results are relevant for numerous different propagation scenarios including microwave and radiowave propagation through various regions of the atmosphere.

math.NA↗

arXiv2017

Stochastic differential equation model for spontaneous emission and carrier noise in semiconductor lasers

We present a new stochastic differential equation model for the spontaneous emission noise and carrier noise in semiconductor lasers. The correlations between these two types of noise have often been neglected in recent studies of the effects of the noise on the laser dynamics. However, the classic results of Henry show that the intensity noise and the carrier noise are strongly negatively correlated. Our model demonstrates how to properly account for these correlations since the corresponding diffusion coefficients agree exactly with those derived by Henry. We show that in fact in the correct model the spontaneous emission noise and the carrier noise are driven by the same Wiener processes. Furthermore, we demonstrate that the nonzero correlation time of the physical noise affects the mean dynamics of both the electric field amplitude and the carrier number. We show that these are systematic corrections that can be described by additional drift terms in the model.

physics.app-ph↗

arXiv2015

Time-averages of Fast Oscillatory Systems in Three-dimensional Geophysical Fluid Dynamics and Electromagnetic Effects

Time-averages are common observables in analysis of experimental data and numerical simulations of physical systems. We will investigate, from the angle of partial differential equation analysis, some oscillatory geophysical fluid dynamics in three dimensions: Navier-Stokes equations in a fast rotating, spherical shell, and Magnetohydrodynamics subject to strong Coriolis and Lorentz forces. Upon averaging their oscillatory solutions in time, interesting patterns such as zonal flows can emerge. More rigorously, we will prove that, when the restoring forces are strong enough, time-averaged solutions stay close to the null spaces of the wave operators, whereas the solutions themselves can be arbitrarily far away from these subspaces.

math.AP↗

arXiv2013

On the Problem of Electromagnetic-Field Quantization

We consider the radiation field operators in a cavity with varying dielectric medium in terms of solutions of Heisenberg's equations of motion for the most general one-dimensional quadratic Hamiltonian. Explicit solutions of these equations are obtained and applications to the radiation field quantization, including randomly varying media, are briefly discussed.

math-ph↗

arXiv2012

Wave Packets and Coherent Structures for Nonlinear Schroedinger equations in Variable Nonuniform Media

We determine conditions under which a generic gauge invariant nonautonomous and inhomogeneous nonlinear partial differential equation in the two-dimensional space-time continuum can be transform into standard autonomous forms. In addition to the nonlinear Schroedinger equation, important examples include the derivative nonlinear Schroedinger equation, the quintic complex Ginzburg-Landau equation, and the Gerdjikov-Ivanov equation. This approach provides a mathematical description of nonstationary media supporting unidimensional signal propagation and/or total field trapping. In particular, we study self-similar and nonspreading wave packets for Schroedinger equations. Some other important coherent structures are also analyzed and applications to nonlinear waves in inhomogeneous media such as atmospheric plasma, fiber optics, hydrodynamics, and Bose-Einstein condensation are discussed.

math-ph↗

arXiv2011

Euler equations on a fast rotating sphere --- time-averages and zonal flows

Motivated by recent studies in geophysical and planetary sciences, we investigate the PDE-analytical aspects of time-averages for barotropic, inviscid flows on a fast rotating sphere $S^2$. Of particular interests are the incompressible Euler equations. We prove that the finite-time-average of the solution stays close to a subspace of \emph{longitude-independent zonal flows}. The intial data can be arbitrarily far away from this subspace. Meridional variation of the Coriolis parameter underlies this phenomenon. Our proofs use Riemannian geometric tools, in particular the Hodge Theory.

math.AP↗

arXiv2007

Bursting Dynamics of the 3D Euler Equations in Cylindrical Domains

A class of three-dimensional initial data characterized by uniformly large vorticity is considered for the Euler equations of incompressible fluids. The fast singular oscillating limits of the Euler equations are studied for parametrically resonant cylinders. Resonances of fast swirling Beltrami waves deplete the Euler nonlinearity. The resonant Euler equations are systems of three-dimensional rigid body equations, coupled or not. Some cases of these resonant systems have homoclinic cycles, and orbits in the vicinity of these homoclinic cycles lead to bursts of the Euler solution measured in Sobolev norms of order higher than that corresponding to the enstrophy.

math.AP↗