Crank-Nicolson Scheme for Solving the Modified Nonlinear Schrodinger Equation

Purpose The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme. Design/methodology/approach The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities. Findings The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients. Originality/value The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.

Theoretical and Experimetal Hints to a Flux of Doppler-Transformation-Energies during processes with Energy Conservation

In a microscopic model of the photoelectric effect it becomes clear that the conservation of energy is exclusively determined by Doppler shift processes, i.e., the whole energy of the pho-ton vanishes by means of Doppler redshifts. Accordingly, if a photon is generated, the energy is won by Doppler blueshifts. This is supposed to be valid for all processes with energy con-servation. An experiment is carried out to make this Doppler energy flow visible by means of interactions with probes. The result of this experiment is that a weak force is measurable in the vicinity of processes with energy conservation. With the aid of a twisted rubber driven low-power device ( ), accelerations of about 10-6 m/s2 are measurable. In the close vicinity of the device, accelerations with values up to 10-3 m/s2 can be concluded. The conse-quences which result from this force are discussed. Keywords: Compton-Effect, Doppler-Effect, energy- momentum conservation, flywheel

4-Index theory of gravity and its relation with the violation of the energy–momentum conservation law

Recently, a 4-index generalization of the Einstein theory has been proposed by Moulin [F. Moulin, Eur. Phys. J. C 77, 878 (2017)]. Using this method, we find the most general 2-index field equations derivable from the Einstein–Hilbert action. The application of Newtonian limit, the role of gravitational coupling constant and the effects of the properties of ordinary energy–momentum tensor in obtaining a 4-index gravity theory have been studied. We also address the results of building Weyl free 4-index gravity theory. Our study displays that both the Einstein and Rastall theories can be obtained as the subclasses of a 4-index gravity theory which shows the power of 4-index method in unifying various gravitational theories. It is also obtained that the violation of the energy–momentum conservation law may be allowed in 4-index gravity theory, and moreover, the contraction of 4-index theory generally admits a non-minimal coupling between geometry and matter field in the Rastall way. This study also shows that, unlike the Einstein case, the gravitational coupling constant of 4-index Rastall theory generally differs from that of the ordinary 2-index Rastall theory.

The energy–momentum conservation law in two-particle system for twist-deformed Galilei Hopf algebras

In this paper, we discuss the energy–momentum conservation principle for two-particle system in the case of canonically and Lie-algebraically twist-deformed Galilei Hopf algebra. Particularly, we provide consistency with the coproducts energy and momentum addition law as well as its symmetric with respect to the exchange of particles counterpart. Besides, we show that the vanishing of total four momentum for two Lie-algebraically deformed kinematical models leads to the discrete values of energies and momenta only in the case of the symmetrized addition rules.

Energy Definition and Dark Energy: A Thermodynamic Analysis

Accepting the Komar mass definition of a source with energy-momentum tensor Tμν and using the thermodynamic pressure definition, we find a relaxed energy-momentum conservation law. Thereinafter, we study some cosmological consequences of the obtained energy-momentum conservation law. It has been found out that the dark sectors of cosmos are unifiable into one cosmic fluid in our setup. While this cosmic fluid impels the universe to enter an accelerated expansion phase, it may even show a baryonic behavior by itself during the cosmos evolution. Indeed, in this manner, while Tμν behaves baryonically, a part of it, namely, Tμν(e) which is satisfying the ordinary energy-momentum conservation law, is responsible for the current accelerated expansion.

Thermodynamics of black holes in Rastall gravity

A promising theory in modifying general relativity (GR) by violating the ordinary energy–momentum conservation law in curved spacetime is the Rastall theory of gravity. In this theory, geometry and matter fields are coupled to each other in a nonminimal way. Here, we study thermodynamic properties of some black hole (BH) solutions in this framework, and compare our results with those of GR. We demonstrate how the presence of these matter sources amplifies the effects caused by the Rastall parameter in thermodynamic quantities. Our investigation also shows that BHs with radius smaller than a certain amount ([Formula: see text]) have negative heat capacity in the Rastall framework. In fact, it is a lower bound for the possible values of horizon radius satisfied by the stable BHs.

Gravitation and energy-momentum conservation in nonsingular general relativity

Forced to be flat when nonsingularly described by the equationsof general relativity in a synchronous frame, a region outsidematter sources is recognized to fail in defining space-time curvature consistently with the material nature of the \mbox{Lorentz} transformation.The curvature then extends to such a region only when there is a continuousbackground medium there. Arising from vacuum decay in the universe, as any matter, this medium is thus capable of avoiding the singularity of gravitational collapseas well. The theory of general relativity is then suggested to stem from sucha formulation of the generalized postulate of relativity as also includes thecovariance of energy-momentum conservation for a macroscopically continuousmaterial system, namely the universe. Implying identity between gravitation andinertia, this formulation does not need the principle of equivalence as aseparate postulate. The \mbox{Einstein} tensor is thus interpretable asstanding for the energy-momentum of the gravitational field. In termsof the background medium, its small-amplitude approximation underlainby matter dynamics and phase transitions also describes what isviewable as gravitational waves. In the framework of such amacroscopic interpretation, gravitation ought to beirrelevant to purely microscopic interactions.