Quaternionic formulation of the classical fields

Long time behavior of solutions of nonlinear classical field equations: the example of NLS defocusing

We discuss two aspects of the behavior of solitary waves in non-integrable classical field theories in one spatial dimension and contrast this behavior with that observed in closely related integrable theories. First, we examine the nature of kink/anti-kink interactions in the φ 4 theory, described by the classical field equation $$ {\phi _{tt}} - {\phi _{xx}} - \phi + {\phi ^3} = 0.$$

By employing the terminology of functional differential calculus, Hamilton-Jacobi theory is extended to apply to classical field equations. It is shown that an asymptotic solution to the Hamilton-Jacobi functional differential equation provides an asymptotic general solution to the associated nonlinear classical field equations.

This work has considered a mathematical model describing the spread of COVID-19 in a given population. The model comprised 5 systems of equations that take into account different classes describing the impact of COVID-19 in a given population. The time differential operator was replaced with three different types of nonlocal operators. These operators are defined as the convolution of variable order fractal differential operators with different kernels including power law, exponential decay law, and Mittag-Leffler functions. We presented the well-poseness of the models for different differential operators that were presented in detail. A novel numerical scheme was used to solve numerically the system and numerical simulations were provided.

A space-time functional formalism for classical field equations

Classical Yang-Mills field equations on Minkowski space with gaugegroup SU(2) are determined. Solutions invariant up to a gauge transformation under a four-dimensional subgroup of the conformal group are evaluated for the Yang-Mills field in the absence of matter. Some properties of the solutions of the equations such as their asymptotic behaviour are obtained.PACS No.: 03.50-z

In the development of quantum mechanics in the 1920s, both matrix mechanics (developed by Born, Heisenberg and Jordon) and wave mechanics (developed by Schrödinger) prevailed. These early attempts corresponded to the quantum mechanics of particles. Matrix mechanics was found to lead directly to the Schrödinger equation, and the Schrödinger equation could be used to derive the alternative problem for matrix mechanics. Later emphasis lay on the development of the dynamics of fields, where the classical field equations were quantized (see, for example, Weinberg). Today, quantum field theory is one of the most successful physical theories ever developed. The symmetry between particle and wave mechanics is exploited herein. One of the important properties of quantum mechanics is that it is linear, leading to some confusion about how to treat the problem of nonlinear classical field equations. In the present paper we address the case of classical nonlinear soliton equations which are exactly integrable in terms of the periodic/quasiperiodic inverse scattering transform. This means that all physical spectral solutions of the soliton equations can be computed exactly for these specific boundary conditions. Unfortunately, such solutions are highly nonlinear, leading to difficulties in solving the associated quantum mechanical problems. Here we find a strategy for developing the quantum mechanical solutions for soliton dynamics. To address this difficulty, we apply a recently derived result for soliton equations, i.e., that all solutions can be written as quasiperiodic Fourier series. This means that soliton equations, in spite of their nonlinear solutions, are perfectly linearizable with quasiperiodic boundary conditions, the topic of finite gap theory, i.e., the inverse scattering transform with periodic/quasiperiodic boundary conditions. We then invoke the result that soliton equations are Hamiltonian, and we are able to show that the generalized coordinates and momenta also have quasiperiodic Fourier series, a generalized linear superposition law, which is valid in the case of nonlinear, integrable classical dynamics and is here extended to quantum mechanics. Hamiltonian dynamics with the quasiperiodicity of inverse scattering theory thus leads to matrix mechanics. This completes the main theme of our paper, i.e., that classical, nonlinear soliton field equations, linearizable with quasiperiodic Fourier series, can always be quantized in terms of matrix mechanics. Thus, the solitons and their nonlinear interactions are given an explicit description in quantum mechanics. Future work will be formulated in terms of the associated Schrödinger equation.

The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations.

Relativistic solitons and superluminal signals

This work presents a predictive and today's most complete model for electrostatic gating of nanofluidic channels. The model is based on the classical mean field equations and accounts for the surface reactivity. The model is successfully verified against previously published experimental data [1]. The fabrication concept of a novel device is introduced and its predicted performances are discussed with the help of the presented model. A ~40x performance improvement, in terms of tunability of the ionic conductivity in the nanochannel, in comparison to the result presented in [1], is expected.

Exact solutions for a generalized Higgs equation

Under normal circumstances most members of the general relativity community focus almost exclusively on the local properties of spacetime, such as the locally Euclidean structure of the manifold and the Lorentzian signature of the metric tensor. When combined with the classical Einstein field equations this gives an extremely successful empirical model of classical gravity and classical matter — at least as long as one does not ask too many awkward questions about global issues, (such as global topology and global causal structure). We feel however that this is a tactical error — even without invoking full-fledged “quantum gravity” we know that the standard model of particle physics is also an extremely good representation of some parts of empirical reality; and we had better be able to carry over all the good features of the standard model of particle physics — at least into the realm of semi-classical quantum gravity. Doing so gives us some interesting global features that spacetime should possess: On physical grounds spacetime should be space-orientable, time-orientable, and spacetime-orientable, and it should possess a globally defined tetrad (vierbein, or in general a globally defined vielbein/[Formula: see text]-bein). So on physical grounds spacetime should be parallelizable. This strongly suggests that the metric is not the fundamental physical quantity; a very good case can be made for the tetrad being more fundamental than the metric. Furthermore, a globally-defined “almost complex structure” is almost unavoidable. Ideas along these lines have previously been mooted, but much is buried in the pre- arXiv literature and is either forgotten or inaccessible. We shall revisit these ideas taking a perspective very much based on empirical physical observation.

Hyperbolic non-conserved phase field equations

Quantitative 3D phase field modelling of solidification using next-generation adaptive mesh refinement

We study in detail the classical Yang-Mills field equations in the presence of static external sources. Their formulation as an initial-value problem in the ${A}_{0}=0$ gauge provides us with a powerful tool for determining the existence of new solutions. In the case of point sources, the only static solutions known so far are the various Coulomb solutions which we classify according to their total energy and isospin. In the case of a localized but extended source, there are, besides the well-known Coulomb solution, two new types of solutions: the "magnetic dipole" solution which has the long-range behavior of a magnetic dipole field and which has lower energy than the Coulomb solution when the total external charge is large enough, and the "total screening" solution which has no long-range field strengths at all and which can have an arbitrarily low energy. We present a detailed study of these new solutions.