scholarly journals Space-Time Coupling: Current Concept and Two Examples from Ultrafast Optics Studied Using Exact Solution of EM Equations

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 529
Author(s):  
Nikolay L. Popov ◽  
Alexander V. Vinogradov
Keyword(s):  
Wave Equations ◽  
Coherent Control ◽  
Ultrafast Optics ◽  
Spectral Width ◽  
Space Time ◽  
Current Approach ◽  
Unipolar Pulse ◽  
Number Of Cycles ◽  
Control Of Quantum Systems ◽  
Coupling Current

Current approach to space-time coupling (STC) phenomena is given together with a complementary version of the STC concept that emphasizes the finiteness of the energy of the considered pulses. Manifestations of STC are discussed in the framework of the simplest exact localized solution of Maxwell’s equations, exhibiting a “collapsing shell”. It falls onto the center, continuously deforming, and then, having reached maximum compression, expands back without losing energy. Analytical solutions describing this process enable to fully characterize the field in space-time. It allowed to express energy density in the center of collapse in the terms of total pulse energy, frequency and spectral width in the far zone. The change of the pulse shape while travelling from one point to another is important for coherent control of quantum systems. We considered the excitation of a two-level system located in the center of the collapsing EM (electromagnetic) pulse. The result is again expressed through the parameters of the incident pulse. This study showed that as it propagates, a unipolar pulse can turn into a bipolar one, and in the case of measuring the excitation efficiency, we can judge which of these two pulses we are dealing with. The obtained results have no limitation on the number of cycles in a pulse. Our work confirms the productivity of using exact solutions of EM wave equations for describing the phenomena associated with STC effects. This is facilitated by rapid progress in the search for new types of such solutions.

scholarly journals Global existence and decay estimates for nonlinear wave equations with space-time dependent dissipative term

2015 ◽  
Vol 12 (02) ◽  
pp. 249-276
Author(s):  
Tomonari Watanabe
Keyword(s):  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Space Time ◽  
Time Dependent ◽  
Nonlinear Wave Equations ◽  
Energy Estimates ◽  
Space Region ◽  
Decay Estimates ◽  
Dissipative Term

We study the global existence and the derivation of decay estimates for nonlinear wave equations with a space-time dependent dissipative term posed in an exterior domain. The linear dissipative effect may vanish in a compact space region and, moreover, the nonlinear terms need not be in a divergence form. In order to establish higher-order energy estimates, we introduce an argument based on a suitable rescaling. The proposed method is useful to control certain derivatives of the dissipation coefficient.

scholarly journals POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 81-141 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
SZYMON ŁȨSKI
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Cauchy Problems ◽  
Time Dimension ◽  
Wave Maps ◽  
Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

scholarly journals PROBABILITY IN RELATIVISTIC QUANTUM MECHANICS AND FOLIATION OF SPACE–TIME

2007 ◽  
Vol 22 (32) ◽  
pp. 6243-6251 ◽  
Author(s):  
HRVOJE NIKOLIĆ
Keyword(s):  
Quantum Mechanics ◽  
Wave Equations ◽  
Integral Curve ◽  
Relativistic Quantum ◽  
Space Time ◽  
Relativistic Quantum Mechanics ◽  
Relativistic Wave ◽  
Probability Densities ◽  
The Many ◽  
Conserved Currents

The conserved probability densities (attributed to the conserved currents derived from relativistic wave equations) should be nonnegative and the integral of them over an entire hypersurface should be equal to one. To satisfy these requirements in a covariant manner, the foliation of space–time must be such that each integral curve of the current crosses each hypersurface of the foliation once and only once. In some cases, it is necessary to use hypersurfaces that are not spacelike everywhere. The generalization to the many-particle case is also possible.

scholarly journals On the Fractional Wave Equation

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 874
Author(s):  
Francesco Iafrate ◽  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Wave Equation ◽  
Wave Equations ◽  
Stable Process ◽  
Space Time ◽  
Probabilistic Interpretation ◽  
Symmetric Stable Process ◽  
Fractional Wave Equation ◽  
Fractional Wave Equations ◽  
Time Fractional Wave Equation

In this paper we study the time-fractional wave equation of order 1 < ν < 2 and give a probabilistic interpretation of its solution. In the case 0 < ν < 1 , d = 1 , the solution can be interpreted as a time-changed Brownian motion, while for 1 < ν < 2 it coincides with the density of a symmetric stable process of order 2 / ν . We give here an interpretation of the fractional wave equation for d > 1 in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for ν > 2 and also at space-time fractional wave equations.

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