Following a hypothesis by Marciak-Kozlowska, 2011, we consider one-dimensional Schumann wave transfer phenomena. Numerical solution of that equation was obtained by the help of Mathematica.

It has been long known that a year after Schrödinger published his equation, Madelung also published a hydrodynamics version of Schrödinger equation. Quantum diffusion is studied via dissipative Madelung hydrodynamics. Initially the wave packet spreads ballistically, than passes for an instant through normal diffusion and later tends asymptotically to a sub‐diffusive law. In this paper we will review two different approaches, including Madelung hydrodynamics and also Bohm potential. Madelung formulation leads to diffusion interpretation, which after a generalization yields to (...) Ermakov equation. Since Ermakov equation cannot be solved analytically, then we try to find out its solution with Mathematica package. It is our hope that these methods can be verified and compared with experimental data. But we admit that more researches are needed to fill all the missing details. (shrink)

We give a rough statement of the main result. Let D be a compact subset of ℝ3× ℝ. The propagation u of a wave can be noncomputable in any neighborhood of any point of D even though the initial conditions which determine the wave propagation uniquely are computable. A precise statement of the result appears below.

This study presents a modified simplest equation method to investigate some real and exact solutions of conformable time fractional Benjamin-Bona-Mahony equation and Chan-Hilliard equation. We use traveling wave transformation to obtain the results in the form of series solution. Some calculations are performed through Mathematica software to analyze the accuracy of this approach. Graphical representations are reported for more significant results at different fractional-order which demonstrates that this approach is very simple, adequate, and legitimate.

In recent years, there are a number of proposals to consider collision-based soliton computer based on certain chemical reactions, namely Belousov-Zhabotinsky reaction, which leads to soliton solutions of coupled Nonlinear Schroedinger equations. They are called Manakov System. But it seems to us that such a soliton computer model can also be based on solitary wave solution of Maxwell-Dirac equation, which reduces to Choquard equation. And soliton solution of Choquard equation has been investigated by many researchers, therefore it (...) seems more profound from physics perspective. However, we consider both schemes of soliton computer are equally possible. More researches are needed to verify our proposition. (shrink)

Starting from a nonlinear relativistic Klein-Gordon equation derived from the stochastic interpretation of quantum mechanics (proposed by Bohm-Vigier, (1) Nelson, (2) de Broglie, (3) Guerra et al. (4) ), one can construct joint wave and particle, soliton-like solutions, which follow the average de Broglie-Bohm (5) real trajectories associated with linear solutions of the usual Schrödinger and Klein-Gordon equations.

We apply a method analogous to the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold, using a method which identifies the symplectic structure of the corresponding mechanics, to the five dimensional generalization of Maxwell theory required by the gauge invariance of Stueckelberg's covariant classical and quantum dynamics. In this way, we demonstrate, in the eikonal approximation, the existence of geodesic motion for the flow of mass (...) in a four dimensional pseudo-Riemannian manifold. These results provide a foundation for the geometrical optics of the five dimensional radiation theory and establish a model in which there is mass flow along geodesics. We then discuss the interesting case of relativistic quantum theory in an anisotropic medium as well. In this case the eikonal approximation to the relativistic quantum mechanical current coincides with the geodesic flow governed by the pseudo-Riemannian metric obtained from the eikonal approximation to solutions of the Stueckelberg–Schrödinger equation. The locally symplectic structure which emerges is that of a generally covariant form of Stueckelberg's mechanics on this manifold. (shrink)

In this paper, we study the diversity of interaction solutions of a shallow water wave equation, the generalized Hirota–Satsuma–Ito equation. Using the Hirota direct method, we establish a general theory for the diversity of interaction solutions, which can be applied to generate many important solutions, such as lumps and lump-soliton solutions. This is an interesting feature of this research. In addition, we prove this new model is integrable in Painlevé sense. Finally, the diversity of interactive (...) wave solutions of the gHSI is graphically displayed by selecting specific parameters. All the obtained results can be applied to the research of fluid dynamics. (shrink)

Traveling-wave solutions of the standard and compound form of Korteweg–de Vries–Burgers equations are found using factorizations of the corresponding reduced ordinary differential equations. The procedure leads to solutions of Bernoulli equations of non-linearity 3/2 and 2 (Riccati), respectively. Introducing the initial conditions through an imaginary phase in the traveling coordinate, we obtain all the solutions previously reported, some of them being corrected here, and showing, at the same time, the presence of interesting details (...) of these solitary waves that have been overlooked before this investigation. (shrink)

The matrix notation of paper I is extended to include first-rank spinors expressed as two-component spin-vectors. Well-known two-component and four-component spinor equations are expressed in this notation. In addition, it is shown how other covariant wave equations can easily be invented. A certain nonlinear equation is found to have only positive-energy solutions for particles and antiparticles.

The correspondence principle states that the quantum system will approach the classical system in high quantum numbers. Indeed, the average of the quantum probability density distribution reflects a classical-like distribution. However, the probability of finding a particle at the node of the wave function is zero. This condition is recognized as the nodal issue. In this paper, we propose a solution for this issue by means of complex quantum random trajectories, which are obtained by solving the stochastic differential equation (...) derived from the optimal guidance law. It turns out that point set A, which is formed by the intersections of complex random trajectories with the real axis, can represent the quantum mechanical compatible distribution of the quantum harmonic oscillator system. Meanwhile, the projections of complex quantum random trajectories on the real axis form point set B that gives a spatial distribution without the appearance of nodes, and approaches the classical compatible distribution in high quantum numbers. Furthermore, the statistical distribution of point set B is verified by the solution of the Fokker–Planck equation. (shrink)

We aim to construct exact and explicit solutions to a generalized Bogoyavlensky-Konopelchenko equation through the Maple computer algebra system. The considered nonlinear equation is transformed into a Hirota bilinear form, and symbolic computations are made for solving both the nonlinear equation and the corresponding bilinear equation. A few classes of exact and explicit solutions are generated from different ansätze on solution forms, including traveling wave solutions, two-wave solutions, and polynomial solutions.

This paper starts from a nonlinear fermion field equation of motion with a strongly coupled self-interaction. Nonperturbative quark solutions of the equation of motion are constructed in terms of a Reggeized infinite component free spinor field. Such a field carries a family of strongly interacting unstable compounds lying on a Regge locus in the analytically continued quark spin. Such a quark field is naturally confined and also possesses the property of asymptotic freedom. Furthermore, the particular field self-regularizes the interactions (...) and naturally breaks the chiral invariance of the equation of motion. We show why and how the existence of such a strongly coupled solution and its particle-family, wave duality forces a change in the field equation of motion such that it conserves C, P, T, although its individual interaction terms are of V-A and thus C, P nonconserving type. (shrink)

The aesthetic field equations do not resemble the wave equation, nor was the motivation behind them the wave equation. Nevertheless, we show that there exists a solution to the field equations that satisfies the wave equation. Integrability is also satisfied by this solution. Previously we showed that the Aesthetic Field Equations have particle solutions. Now we see that the equations also have sinusoidal solutions.

The special and general relativity theories are used to demonstrate that the velocity of an unradiative particle in a Schwarzschild metric background, and in an electrostatic field, is the group velocity of a wave that we call a “particle wave,” which is a monochromatic solution of a standard equation of wave motion and possesses the following properties. It generalizes the de Broglie wave. The rays of a particle wave are the possible particle trajectories, and the (...) motion equation of a particle can be obtained from the ray equation. The standing particle wave equation generalizes the Schrödinger equation of wave amplitudes. The particle wave motion equation generalizes the Klein–Gordon equation; this result enables us to analyze the essence of the particle wave frequency. The equation of the eikonal of a particle wave generalizes the Hamilton–Jacobi equation; this result enables us to deduce the general expression for the linear momentum. The Heisenberg uncertainty relation expresses the diffraction of the particle wave, and the uncertainty relation connecting the particle instant of presence and energy results from the fact that the group velocity of the particle wave is the particle velocity. A single classical particle may be considered as constituted of geometrical particle wave; reciprocally, a geometrical particle wave may be considered as constituted of classical particles. The expression for a particle wave and the motion equation of the particle wave remain valid when the particle mass is zero. In that case, the particle is a photon, the particle wave is a component a classical electromagnetic wave that is embedded in a Schwarzschild metric background, and the motion equation of the wave particle is the motion equation of an electromagnetic wave in a Schwarzschild metric background. It follows that a particle wave possesses the same physical reality as a classical electromagnetic wave. This last result and the fact that the particle velocity is the group velocity of its wave are in accordance with the opinions of de Broglie and of Schrödinger. We extend these results to the particle subjected to any static field of forces in any gravitational metric background. Therefore we have achieved a synthesis of undulatory mechanics, classical electromagnetism, and gravitation for the case where the field of forces and the gravitational metric background are static, and this synthesis is based only on special and general relativity. (shrink)

A rigorous ab initio derivation of the (square of) Dirac’s equation for a particle with spin is presented. The Lagrangian of the classical relativistic spherical top is modified so to render it invariant with respect conformal changes of the metric of the top configuration space. The conformal invariance is achieved by replacing the particle mass in the Lagrangian with the conformal Weyl scalar curvature. The Hamilton-Jacobi equation for the particle is found to be linearized, exactly and in closed form, by (...) an ansatz solution that can be straightforwardly interpreted as the “quantum wave function” of the 4-spinor solution of Dirac’s equation. All quantum features arise from the subtle interplay between the conformal curvature acting on the particle as a potential and the particle motion which affects the geometric “pre-potential” associated to the conformal curvature itself. The theory, carried out here by assuming a Minkowski metric, can be easily extended to arbitrary space-time Riemann metric, e.g. the one adopted in the context of General Relativity. This novel theoretical scenario appears to be of general application and is expected to open a promising perspective in the modern endeavor aimed at the unification of the natural forces with gravitation. (shrink)

This paper exploits the modified simple equation and dynamical system schemes to integrate the Klein–Gordon model amid quadratic nonlinearity arising in nonlinear optics, quantum theories, and solid state physics. By implementing the modified simple equation technique, we develop some disguise adaptation of analytical solutions in terms of hyperbolic, exponential, and trigonometric functions with some special parameters. We apply the dynamical system to bifurcate the model and draw distinct phase portraits on unlike parametric constraints. Following each orbit of all phase (...) portraits, we originate bounded and unbounded solitary, periodic, and periodic rogue-type wave solutions of the KG model. These two schemes extract widespread classes of solitary, periodic, and periodic rogue-type wave solutions for the KG model jointly due to restrictions on parameters. We also analyze the effect of parameters on the obtained wave solutions and discuss why and when it changes its nature. We illustrate some dynamical features of the acquired solutions via the 3D, 2D, and contour graphics. (shrink)

In this study, some new hypotheses and techniques are presented to obtain some new analytical solutions to the generalized Kawahara equation. As a particular case, some traveling wave solutions to both Kawahara equation and modified Kawahara equation are derived in detail. Periodic and soliton solutions to this family are obtained. The periodic solutions are expressed in terms of Weierstrass elliptic functions and Jacobian elliptic functions. For KE, some direct and indirect approaches are carried out to (...) derive the periodic and localized solutions. For mKE, two different hypotheses in the form of WSEFs are used to derive the periodic and localized solutions. Also, the cnoidal wave solutions in the form of JEFs are obtained. As a realistic physical application, the solutions obtained can be dedicated to studying many nonlinear waves that propagate in plasma. (shrink)

The physical meaning of the particularly simple non-degenerate supermetric, introduced in the previous part by the authors, is elucidated and the possible connection with processes of topological origin in high energy physics is analyzed and discussed. New possible mechanism of the localization of the fields in a particular sector of the supermanifold is proposed and the similarity and differences with a 5-dimensional warped model are shown. The relation with gauge theories of supergravity based in the OSP(1/4) group is explicitly given (...) and the possible original action is presented. We also show that in this non-degenerate super-model the physic states, in contrast with the basic states, are observables and can be interpreted as tomographic projections or generalized representations of operators belonging to the metaplectic group Mp(2). The advantage of geometrical formulations based on non-degenerate super-manifolds over degenerate ones is pointed out and the description and the analysis of some interesting aspects of the simplest Riemannian superspaces are presented from the point of view of the possible vacuum solutions. (shrink)

Multi-time wave functions are wave functions for multi-particle quantum systems that involve several time variables. In this paper we contrast them with solutions of wave equations on a space–time with multiple timelike dimensions, i.e., on a pseudo-Riemannian manifold whose metric has signature such as \ or \, instead of \. Despite the superficial similarity, the two behave very differently: whereas wave equations in multiple timelike dimensions are typically mathematically ill-posed and presumably unphysical, relevant (...) Schrödinger equations for multi-time wave functions possess for every initial datum a unique solution on the spacelike configurations and form a natural covariant representation of quantum states. (shrink)

Dirac's equation is reviewed and found to be based on nonrelativistic ideas of probability. A 4-space formulation is proposed that is completely Lorentzinvariant, using probability distributions in space-time with the particle's proper time as a parameter for the evolution of the wave function. This leads to a new wave equation which implies that the proper mass of a particle is an observable, and is sharp only in stationary states. The model has a built-in arrow of time, which is (...) associated with a restriction to positive-energy solutions. The usual solution for a Coulomb field is retained, though it now implies a slightly different charge distribution. The conventional nonstationary solutions become invalid. The new formulation appears to offer a resolution of difficulties that have been associated with Dirac's equation. It also predicts the occurrence of virtual pairs at a level that may be experimentally testable, and suggests a mechanism for self-cancellation of the vacuum energy. (shrink)

The basic idea of quantum mechanics is that the property of any system can be in a state of superposition of various possibilities. This state of superposition is also known as wave function and it evolves linearly with time in a deterministic way in accordance with the Schrodinger equation. However, when a measurement is carried out on the system to determine the value of that property, the system instantaneously transforms to one of the eigen states and thus we get (...) only a single value as outcome of the measurement. Quantum measurement problem seeks to find the cause and exact mechanism governing this transformation. In an attempt to solve the above problem, in this paper, we will first define what the wave function represents in real world and will identify the root cause behind the stochastic nature of events. Then, we will develop a model to explain the mechanism of collapse of the quantum mechanical wave function in response to a measurement. In the process of development of model, we will explain Schrodinger cat paradox and will show how Born’s rule for probability becomes a natural consequence of measurement process. (shrink)

This study employs a newly developed methodology called the variational homotopy perturbation transformation method to study fractional acoustic wave equations. The motivation for this study is to extend the variational homotopy perturbation technique to the variational homotopy perturbation transformation technique in the sense of the Yang–Caputo–Fabrizio operator. The suggested method demonstrated a straightforward and accurate technique for investigating fractional-order partial differential equations. The technique’s validity is demonstrated through the use of several illustrative instances. The obtained answers were (...) found to be extremely near to the precise solutions. Additionally, the proposed strategy achieves the best degree of accuracy. Indeed, the current technique can be seen as one of the analytic strategies for solving nonlinear fractional partial differential equations compared to other analytical techniques. (shrink)

In this paper, a generalized -dimensional Calogero–Bogoyavlenskii–Schiff equation is considered. Based on the Hirota bilinear method, three kinds of exact solutions, soliton solution, breather solutions, and lump solutions, are obtained. Breathers can be obtained by choosing suitable parameters on the 2-soliton solution, and lump solutions are constructed via the long wave limit method. Figures are given out to reveal the dynamic characteristics on the presented solutions. Results obtained in this work may be conducive to (...) understanding the propagation of localized waves. (shrink)

In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He’s fractional complex transform. This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method is employed to tackle the nonlinear terms in (...) the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two-scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering. (shrink)

We illustrate, using a simple model, that in the usual formulation the time-component of the Klein–Gordon current is not generally positive definite even if one restricts allowed solutions to those with positive frequencies. Since in de Broglie's theory of particle trajectories the particle follows the current this leads to difficulties of interpretation, with the appearance of trajectories which are closed loops in space-time and velocities not limited from above. We show that at least this pathology can be avoided if (...) one adapts in a covariant form the formulation of relativistic point particle dynamics proposed by Gitman and Tyutin. (shrink)

We present a classical hydrodynamic analog of free relativistic quantum particles inspired by de Broglie’s pilot wave theory and recent developments in hydrodynamic quantum analogs. The proposed model couples a periodically forced Klein–Gordon equation with a nonrelativistic particle dynamics equation. The coupled equations may represent both quantum particles and classical particles driven by the gradients of locally excited Faraday waves. Exact stationary solutions of the coupled system reveal a highly nonlinear mechanism responsible for the self-propulsion of free (...) particles, leading to the onset of unsteady motion. Although the model is essentially nonrelativistic, a stabilizing mechanism for any particle traveling close to the wave signaling speed emerges through the coupling with the wavefield. Consequently, inline particle oscillations comparable to de Broglie’s wavelength are realized through this fully-classical model, suggesting a new classical interpretation for the motion of relativistic quantum particles. (shrink)

The covariant Klein-Gordon equation requires twice the boundary conditions of the Schrödinger equation and does not have an accepted single-particle interpretation. Instead of interpreting its solution as a probability wave determined by an initial boundary condition, this paper considers the possibility that the solutions are determined by both an initial and a final boundary condition. By constructing an invariant joint probability distribution from the size of the solution space, it is shown that the usual measurement probabilities can nearly (...) be recovered in the non-relativistic limit, provided that neither boundary constrains the energy to a precision near ℏ/t 0 (where t 0 is the time duration between the boundary conditions). Otherwise, deviations from standard quantum mechanics are predicted. (shrink)

We investigate the problem of “wave packet reduction” in quantum mechanics by solving the Schrödinger equation for a system composed of a model measuring apparatusM interacting with a microscopic objects. The “instrument” is intended to be somewhat more realistic than others previously proposed, but at the same time still simple enough to lead to an explicit solution for the time-dependent density matrix. It turns out that,practically, everything happens as if the wave packet reduction had occurred. This is a (...) consequence of the fact that the apparatus is made up of a very large number of microsystems interacting withs. More precisely, our model shows that the “macroscopic size” of a measuring apparatus can lead by itself to a density matrix for the systemM + s which is physically equivalent to the density matrix of a statistical mixture corresponding to the reduced wave packet. (shrink)

In QED, an external electromagnetic field can be accounted for non-perturbatively by replacing the causal propagators used in Feynman diagram calculations with Green’s functions for the Dirac equation under the external field. If the external field destabilises the vacuum, then it is a difficult problem to determine which Green’s function is appropriate, and multiple approaches have been developed in the literature whose equivalence, in many cases, is not clear. In this paper, we demonstrate for a broad class of external fields (...) that includes all that act for a finite time, that four Green’s functions used in the literature are equivalent: Schwinger’s “proper-time” propagator; the “causal propagator” used in the “Bogoliubov transformation” method based on the canonical quantization of the field operator; and two defined using analytic continuation from complexified parameters. To do so, we formulate Schwinger’s “proper-time quantum mechanics" as a Schrödinger wave mechanics, and use this to re-derive Schwinger’s expression for the propagator as a statement relating solutions of the inhomogeneous Dirac equation to those of the inhomogeneous “proper-time Dirac equation”. This is done by constructing direct integral spectral decompositions of the Hamiltonians of both equations, and deriving a form for solutions of the inhomogeneous Dirac equation in terms of these decompositions. We then show that all four propagators return solutions of the inhomogeneous Dirac equation that satisfy the same boundary condition, which under a physically reasonable assumption is sufficient to specify the solution uniquely. (shrink)

It has been known for long time that the cosmic sound wave was there since the early epoch of the Universe. Signatures of its existence are abound. However, such a sound wave model of cosmology is rarely developed fully into a complete framework. This paper can be considered as our second attempt towards such a complete description of the Universe based on soliton wave solution of cosmological KdV equation. Then we advance further this KdV equation by virtue (...) of Cellular Automaton method to solve the PDEs. We submit wholeheartedly Robert Kurucz’s hypothesis that Big Bang should be replaced with a finite cellular automaton universe with no expansion. Nonetheless, we are fully aware that our model is far from being complete, but it appears the proposed cellular automaton model of the Universe is very close in spirit to what Konrad Zuse envisaged long time ago. It is our hope that the new proposed method can be verified with observation data. But we admit that our model is still in its infancy, more researches are needed to fill all the missing details. (shrink)

In classical electrodynamics, electromagnetic effects are calculated from solution of wave equation formed by combination of four Maxwell’s equations. However, along with retarded solution, this wave equation admits advanced solution in which case the effect happens before the cause. So, to preserve causality in natural events, the retarded solution is intentionally chosen and the advance part is just ignored. But, an equation or method cannot be called fundamental if it admits a wrong result (that violates principle of (...) causality) in addition to the correct result. Since it is the Maxwell’s form of equations that gives birth to this acausal advanced potential, we rewrite these equations in a different form using the recent theory of reaction at a distance (Biswaranjan Dikshit, Physics essays, 24(1), 4-9, 2011) so that the process of calculation does not generate any advanced effects. Thus, the long-standing causality problem in electrodynamics is solved. (shrink)

In this paper, it is presented a historical account of the formulation of the quantum relativistic wave equation of an electron – the Dirac equation, issues regarding its interpretation that arose from the very beginning, and the later formulation of this equation in relation to a quantized electron-positron field, which implies a new interpretation. The way in which solutions obtained under each interpretation of the equation relate to one another is also considered for the simple case of hydrogen-like (...) problems. (shrink)

It is shown that the wave equation of anN-body problem can be transformed into a system of “hydrodynamical equations” in a3N-dimensional space. The projections of the hydrodynamical variables in three-dimensional space do not obey strict equations of motion. This is shown to be connected with the fact that the mathematically possible solutions of the wave equations are much more numerous than the states of the system that are usually realized in nature. It is pointed (...) out that the many-body wave equation has to be supplemented with thermodynamic principles, so as to obtain a satisfactory description of real phenomena. (shrink)

We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs to solutions of Tx = u ? Here we consider the case when T is a bounded linear mapping between Hilbert spaces. We are in particular interested in computing the generalized inverse T†u, which is the standard concept of solution in the theory of inverse problems. Typically, T† is discontinuous and hence no computable mapping. However, we will (...) use effective versions of theorems from the theory of regularization to show that the mapping ↦ T†u is computable. We then go on to study the computability of average-case solutions with respect to Gaussian measures which have been considered in information based complexity. Here, T† is considered as an element of an L2-space. We define suitable representations for such spaces and use the results from the first part of the paper to show that ↦ T† is computable. (shrink)

To construct fractional rogue waves, this paper first introduces a conformable fractional partial derivative. Based on the conformable fractional partial derivative and its properties, a fractional Schrödinger equation with Lax integrability is then derived and first- and second-order fractional rogue wave solutions of which are finally obtained. The obtained fractional rogue wave solutions possess translational coordination, providing, to some extent, the degree of freedom to adjust the position of the rogue waves on the coordinate plane. It (...) is shown that the obtained first- and second-order fractional rogue wave solutions are steeper than those of the corresponding NLS equation with integer-order derivatives. Besides, the time the second-order fractional rogue wave solution undergoes from the beginning to the end is also short. As for asymmetric fractional rogue waves with different backgrounds and amplitudes, this paper puts forward a way to construct them by modifying the obtained first- and second-order fractional rogue wave solutions. (shrink)

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a 0+a 1tan ξ+a (...) 2tan 2 ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method. (shrink)

It is shown that in every gauge the potential of the electromagnetic field in the presence of sources is resolved by an extension of the Helmholtz theorem into a solenoidal component and an irrotational component irrelevant for description of the field. Only irrotational components are affected by gauge transformations; in Coulomb gauge the irrotational component vanishes: the potential is solenoidal. The method of solution of the wave equation by use of positive- and negative-frequency parts is extended to solutions (...) of D'Alembert's equation, and applied to equations satisfied by the potential in Coulomb gauge and the electric and magnetic vectors. Fourier transforms of potentials specifying destruction/creation operators become time dependent in the presence of sources. Our central equation states this time dependence. Frequency parts of Maxwell's equations are obtained. When the retarded potential in Coulomb gauge is resolved into kinetic and dissipative components, the latter is shown to be in radiation gauge. Correspondingly, the energy/stress tensor is resolved into three components; the power/force density, into two: a kinetic and a dissipative component. Work done by the latter component is negative: energy and momentum are dissipated from matter to radiation. Boson quantization conditions are satisfied by the kinetic component of the retarded potential, but commutators of the dissipative component are determined by the current sources. The energy/stress tensor and Hamiltonian of the field in the presence of sources are derived from the classical Lagrangian density. The relation between the Hamiltonian and the energy is shown to agree with the time dependence of the destruction/creation operators in Heisenberg picture. (shrink)

The challenges with modeling the spread of Covid-19 are its power-type growth during the middle stages of the waves with the exponents depending on time, and that the saturation of the waves is mainly due to the protective measures and other restriction mechanisms working in the same direction. The two-phase solution we propose for modeling the total number of detected cases of Covid-19 describes the actual curves for many its waves and in many countries almost with the accuracy of physics (...) laws. Bessel functions play the key role in our approach. The differential equations we obtain are of universal type and can be used in behavioral psychology, invasion ecology (transient processes), etc. The initial transmission rate and the intensity of the restriction mechanisms are the key parameters. This theory provides a convincing explanation of the surprising uniformity of the Covid-19 waves in many places, and can be used for forecasting the epidemic spread. For instance, the early projections for the 3rd wave in the USA appeared sufficiently exact. The Delta-waves (2021) in India, South Africa, UK, and the Netherlands are discussed at the end. (shrink)

In this work, a Clifford algebra approach is used to introduce a charge-current wave structure governed by a Maxwell-like set of equations. A known spinor representation of the electromagnetic field intensities is utilized to recast the equations governing the charge-current densities in a Dirac-like spinor form. Energy-momentum considerations lead to a generalization of the Maxwell electromagnetic symmetric energy-momentum tensor. The generalized tensor includes new terms that represent contributions from the charge-current densities. Stationary spherical modal solutions representing (...) the charge-current densities and the associated self-fields are derived. The use of a Clifford type dependence on time results in a distinct symmetry between the magnetic and electric components. It is shown that, for such spherical modes, the components of the force density deduced from the generalized energy-momentum tensor can vanish under certain conditions. (shrink)

We present the theory, the experimental evidence and fundamental physical consequences concerning the existence of families of undistorted progressive waves (UPWs) of arbitrary speeds 0≤ϑ<∞, which are solutions of the homogeneuous wave equation, the Maxwell equations, and Dirac, Weyl, and Klein-Gordon equations.

In this paper a new trajectory-based representation to non-relativistic quantum mechanics is formulated. This is ahieved by generalizing the notion of Lagrangian path which lies at the heart of the deBroglie-Bohm “ pilot-wave” interpretation. In particular, it is shown that each LP can be replaced with a statistical ensemble formed by an infinite family of stochastic curves, referred to as generalized Lagrangian paths. This permits the introduction of a new parametric representation of the Schrödinger equation, denoted as GLP-parametrization, and (...) of the associated quantum hydrodynamic equations. The remarkable aspect of the GLP approach presented here is that it realizes at the same time also a new solution method for the N-body Schrödinger equation. As an application, Gaussian-like particular solutions for the quantum probability density function are considered, which are proved to be dynamically consistent. For them, the Schrödinger equation is reduced to a single Hamilton–Jacobi evolution equation. Particular solutions of this type are explicitly constructed, which include the case of free particles occurring in 1- or N-body quantum systems as well as the dynamics in the presence of suitable potential forces. In all these cases the initial Gaussian PDFs are shown to be free of the spreading behavior usually ascribed to quantum wave-packets, in that they exhibit the characteristic feature of remaining at all times spatially-localized. (shrink)

An internationally famous physicist and electrical engineer, the author of this text was a pioneer in the investigation of gravitational waves. Joseph Weber's General Relativity and Gravitational Waves offers a classic treatment of the subject. Appropriate for upper-level undergraduates and graduate students, this text remains ever relevant. Brief but thorough in its introduction to the foundations of general relativity, it also examines the elements of Riemannian geometry and tensor calculus applicable to this field. Approximately a quarter of the contents explores (...) theoretical and experimental aspects of gravitational radiation. The final chapter focuses on selected topics related to general relativity, including the equations of motion, unified field theories, Friedman's solution of the cosmological problem, and the Hamiltonian formulation of general relativity. Exercises. Index. (shrink)

The equivalence of Fermat's Last Theorem and the non-existence of solutions of a generalized n th order homogeneous hyperbolic partial differential equation in three dimensions and periodic boundary conditions defined in a cubic lattice is demonstrated for all positive integer, n > 2. For the case n = 2, choosing one variable as time, solutions are identified as either propagating or standing waves. Solutions are found to exist in the corresponding problem in two dimensions.

The phenomena, molecular path in a liquid or a gas, fluctuating price stoke, fission and fusion, quantum field theory, relativistic wave motion, etc., are modeled through the nonlinear time fractional clannish random Walker’s parabolic equation, nonlinear time fractional SharmaTassoOlver equation, and the nonlinear space-time fractional KleinGordon equation. The fractional derivative is described in the sense of conformable derivative. From there, the G ′ / G, 1 / G -expansion method is found to be ensuing, effective, and capable to provide (...) functional solutions to nonlinear models concerning physical and engineering problems. In this study, an extension of the G ′ / G, 1 / G -expansion method has been introduced. This enhancement establishes broad-ranging and adequate fresh solutions. In addition, some existing solutions attainable in the literature also confirm the validity of the suggested extension. We believe that the extension might be added to the literature as a reliable and efficient technique to examine a wide variety of nonlinear fractional systems with parameters including solitary and periodic wave solutions to nonlinear FDEs. (shrink)

In this paper, we propose an ontological interpretation of the wave function in terms of random discontinuous motion of particles. According to this interpretation, the wave function of an N-body quantum system describes the state of random discontinuous motion of N particles, and in particular, the modulus squared of the wave function gives the probability density that the particles appear in every possible group of positions in space. We present three arguments supporting this new interpretation of the (...) wave function. These arguments are mainly based on an analysis of the mass and charge properties of a quantum system. It is realized that the Schroedinger equation, which governs the evolution of a quantum system, contains more information about the system than the wave function of the system, such as the mass and charge properties of the system, which might help understand the ontological meaning of the wave function. Finally, we briefly analyze possible implications of the suggested ontological interpretation of the wave function for the solutions to the measurement problem. (shrink)

An algebraic block-diagonalization of the Dirac Hamiltonian in a time-independent external field reveals a charge-index conservation law which forbids the physical phenomena of the Klein paradox type and guarantees a single-particle nature of the Dirac equation in strong external fields. Simultaneously, the method defines simpler quantum-mechanical objects—paulions and antipaulions, whose 2-component wave functions determine the Dirac electron states through exact operator relations. Based on algebraic symmetry, the presented theory leads to a new understanding of the Dirac equation physics, including (...) new insight into the Dirac measurements and a consistent scheme of relativistic quantum mechanics of electron in the paulion representation. Along with analysis of the mathematical anatomy of the Klein paradox falsity, a complete set of paradox-free eigenfunctions for the Klein problem is obtained and investigated via stationary solutions of the Pauli-like equations with respective paulion Hamiltonians. It is shown that the physically correct Dirac states in the Klein zone are characterized by the total particle reflection from the potential step and satisfy the fundamental charge-index conservation law. (shrink)