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We find and analyze exact solutions to equations with logarithmic nonlinearity. To construct the soltuions, we use the classical group analysis methods, as well as the method of invariant manifolds and the Lagrangian coordinate method. Bibliography: 23 titles. Illustrations: 5 figures.
The model of Euclidean space with imaginary time used in sub-hadron physics uses only part of it since this part is isomorphic to Minkowski space and has the velocity limit 0 ≤ ||v{sub Ei}|| ≤ 1. The model of four-dimensional Euclidean space with real time (Е space), in which 0 ≤ ||v{sub E}|| ≤ ∞ is investigated. The vectors of this space have Е-invariants, equal or analogous to the invariants of Minkowski space. All relations between physical quantities in Е-space, after they are mapped into Minkowski space, satisfy the principles of SRT and are Lorentz-invariant, and the velocity of light corresponds to infinite velocity. Results obtained in the model are different from the physical laws in Minkowski space. Thus, from the model of the Lagrangian mechanics of quarks in a centrally symmetric attractive potential it follows that the energy-mass of a quark decreases with increase of the velocity and is equal to zero for v = ∞. This made it possible to establish the conditions of emission and absorption of gluons by quarks. The effect of emission of gluons by high-energy quarks was discovered experimentally significantly earlier. The model describes for the first time the dynamic coupling of the masses of constituent and current quarks and reveals new possibilities in the study of intrahardon space. The classical trajectory of the oscillation of quarks in protons is described.
Different sizes and shapes of nonmetallic inclusions in a swirling flow submerged entry nozzle (SEN) placed in a new tundish design were investigated by using a Lagrangian particle tracking scheme. The results show that inclusions in the current cylindrical tundish have difficulties remaining in the top tundish region, since a strong rotational steel flow exists in this region. This high rotational flow of 0.7 m/s provides the required momentum for the formation of a strong swirling flow inside the SEN. The results show that inclusions larger than 40 µm were found to deposit to a smaller extent on the SEN wall compared to smaller inclusions. The reason is that these large inclusions have Separation number values larger than 1. Thus, the swirling flow causes these large size inclusions to move toward the SEN center. For the nonspherical inclusions, large size inclusions were found to be deposited on the SEN wall to a larger extent, compared to spherical inclusions. More specifically, the difference of the deposited inclusion number is around 27 pct. Overall, it was found that the swirling flow contains three regions, namely, the isotropic core region, the anisotropic turbulence region and the near-wall region. Therefore, anisotropic turbulent fluctuations should be taken into account when the inclusion motion was tracked in this complex flow. In addition, many inclusions were found to deposit at the SEN inlet region. The plotted velocity distribution shows that the inlet flow is very chaotic. A high turbulent kinetic energy value of around 0.08 m{sup 2}/s{sup 2} exists in this region, and a recirculating flow was also found here. These flow characteristics are harmful since they increase the inclusion transport toward the wall. Therefore, a new design of the SEN inlet should be developed in the future, with the aim to modify the inlet flow so that the inclusion deposition is reduced.
In continuous casting of steel, argon gas is often injected to prevent clogging of the nozzle, but the bubbles affect the flow pattern, and may become entrapped to form defects in the final product. Further, an electromagnetic field is frequently applied to induce a braking effect on the flow field and modify the inclusion transport. In this study, a previously validated GPU-based in-house code CUFLOW is used to investigate the effect of electromagnetic braking on turbulent flow, bubble transport, and capture. Well-resolved large eddy simulations are combined with two-way coupled Lagrangian computations of the bubbles. The drag coefficient on the bubbles is modified to account for the effects of the magnetic field. The distribution of the argon bubbles, capture, and escape rates, are presented and compared with and without the magnetic field. The bubble capture patterns are also compared with results of a previous RANS model as well as with plant measurements.
A transient mathematical model is developed for simulating the bubble-steel-slag-top gas four-phase flow in a bottom-blown argon-stirred ladle with a 70-ton capacity. The Lagrangian discrete phase model (DPM) is used for describing the moving behavior of bubbles in the steel and slag. To observe the formation process of slag eye, the volume of fluid (VOF) model is used to track the interfaces between three incompressible phases: metal/slag, metal/gas, and slag/gas. The complex multiphase turbulent flow induced by bubble-liquid interactions is solved by a large eddy simulation (LES) model. Slag eye area and slag droplet dispersion are investigated under different gas flow rates. The results show that the movement of bubbles, formation and collapse of slag eye, volatility of steel/slag interface and behavior of slag entrapment can be properly predicted in the current model. When the gas flow rate is 300 L/min, the circulation driven by the bubble plume will stir the entire ladle adequately and form a slag eye of the right size. At the same time, it will not cause strong erosion to the ladle wall, and the fluctuation of the interface is of adequate intensity, which will be helpful for improving the desulfurization efficiency; the slag entrapment behavior can also be decreased. Interestingly, with the motion of liquid steel circulation, the collision and coalescence of dispersed slag droplets occur during the floating process in the vicinity of the wall.
This article presents a review of the research into gas stirring in ladle metallurgy carried out over the past few decades. Herein, the physical modeling experiments are divided into four major areas: (1) mixing and homogenization in the ladle; (2) gas bubble formation, transformation, and interactions in the plume zone; (3) inclusion behavior at the steel–slag interface and in the molten steel; and (4) open eye formation. Several industrial trials have also been carried out to optimize gas stirring and open eye formation. Approaches for selecting criteria for scaling to guarantee flow similarity between industrial trials and physical modeling experiments are discussed. To describe the bubble behavior and two-phase plume structure, four main mathematical models have been used in different research fields: (1) the quasi-single-phase model, (2) the volume of fluid (VOF) model, (3) the Eulerian multiphase (E–E) model, and (4) the Eulerian–Lagrangian (E–L) model. In recent years, the E–E model has been used to predict gas stirring conditions in the ladle, and specific models in commercial packages, as well as research codes, have been developed gradually to describe the complex physical and chemical phenomena. Furthermore, the coupling of turbulence models with multiphase models is also discussed. For physical modeling, some general empirical rules have not been analyzed sufficiently. Based on a comparison with the available experimental results, it is found that the mathematical models focusing on the mass transfer phenomenon and inclusion behaviors at the steel-slag interface, vacuum degassing at the gas–liquid interface, dissolution rate of the solid alloy at the liquid–solid interface, and the combination of fluid dynamics and thermodynamics need to be improved further. To describe industrial conditions using mathematical methods and improve numerical modeling, the results of physical modeling experiments and industrial trials must offer satisfactory validations for the improvement of numerical modeling.
Crashworthiness is defined as the capability of a structure to guarantee its occupants safety during a crash event. In the present work, the crashworthiness of a composite wing section, subjected to a bird strike event, has been investigated. Indeed, the mechanical behavior of the impacted wing section has been inspected by means of the FE code Abaqus/Explicit. The impact phenomenon has been numerically simulated with a coupled Eulerian–Lagrangian (CEL) approach. The impacted structure has been modeled considering a Lagrangian formulation, while a Eulerian one has been used to model the bird. Moreover, Eulerian–Lagrangian contacts are used for transfer the loads arising from the impact event to the Lagrangian structure. The proposed numerical model, capable to take into account the matrix and fiber traction and compression damages, uses a 3D finite element approach to better predict the deformations and the damages onset and propagation during the impact event. The numerical results, in terms of structural deformation and damage onset and propagation, have been obtained by considering different impact locations and different impact angles.
In this paper, we study the extended trust region subproblem (eTRS) in which the trust region intersects the Euclidean ball with a single linear inequality constraint. By reformulating the Lagrangian dual of eTRS as a two-parameter linear eigenvalue problem, we state a necessary and sufficient condition for its strong duality in terms of an optimal solution of a linearly constrained bivariate concave maximization problem. This results in an efficient algorithm for solving eTRS of large size whenever the strong duality is detected. Finally, some numerical experiments are given to show the effectiveness of the proposed method.
The present work investigates several models of a single real scalar field, engendering kinetic term of the Dirac–Born– Infeld type. Such theories introduce nonlinearities to the kinetic part of the Lagrangian, which presents a square root restricting the field evolution and including additional powers in derivatives of the scalar field, controlled by a real parameter. In order to obtain topological solutions analytically, we propose a first-order framework that simplifies the equation of motion ensuring solutions that are linearly stable. This is implemented using the deformation method, and we introduce examples presenting two categories of potentials, one having polynomial interactions and the other with nonpolynomial interactions. We also explore how the Dirac–Born–Infeld kinetic term affects the properties of the solutions. In particular, we note that the kinklike solutions are similar to the ones obtained through models with standard kinetic term and canonical potential, but their energy densities and stability potentials vary according to the parameter introduced to control the new models.
This paper shows a new approach to obtain analytical topological defects of a 2D Myers–Pospelov Lagrangian for two scalar fields. Such a Lagrangian presents higher-order kinetic terms, which lead us to equations of motion which are non-trivial to be integrated. Here we describe three possible scenarios for the equations of motion, named by timelike, spacelike and lightlike respectively. We started our investigation with a kink-like traveling wave Ansatz for the free theory, which led us to constraints for the dispersion relations of each scenario. We also introduced a procedure to obtain analytical solutions for the general theory in the three mentioned scenarios. We exemplified the procedure and discussed the behavior of the defect solutions carefully. It is remarkable that the methodology presented in this study led to analytical models, despite the complexity of the equations of motion derived from the 2D Myers–Pospelov Lagrangian. The methodology here tailored can be applied to several Lagrangians with higher-order derivative terms.
Motivated by a geometric decomposition of the vector field associated with the Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation for finite-level open quantum systems, we propose a generalization of the recently introduced contact Hamiltonian systems for the description of dissipative-like dynamical systems in the context of (non-necessarily exact) contact manifolds. In particular, we show how this class of dynamical systems naturally emerges in the context of Lagrangian Mechanics and in the case of nonlinear evolutions on the space of pure states of a finite-level quantum system.
Highlights: •We consider the theoretical description of intense laser pulses propagating through gases. •We derived reduced models using Lagrangian and Hamiltonian formulations. •We obtain models where the atomic electrons are treated classically or quantum-mechanically. •We derive a number of conserved quantities of the reduced models. -- Abstract: We consider the theoretical description of intense laser pulses propagating through gases. Starting from a first-principles description of both the electromagnetic field and the electron motion within the gas atoms, we derive a hierarchy of reduced models. We obtain a parallel set of models, where the atomic electrons are treated classically on the one hand, and quantum-mechanically on the other. By working consistently in either a Lagrangian formulation or a Hamiltonian formulation, we ensure that our reduced models preserve the variational structure of the parent models. Taking advantage of the Hamiltonian formulation, we deduce a number of conserved quantities of the reduced models.
Alternating current (ac) circuits can have electromagnetic edge modes protected by symmetries, analogous to topological band insulators or semimetals. How to make such a topological circuit? This paper illustrates a particular design idea by analyzing a series of topological circuits consisting purely of inductors (L) and capacitors (C) connected to each other by wires to form periodic lattices. All the examples are treated using a unifying approach based on Lagrangians and the dynamical H-matrix. First, the building blocks and permutation wiring are introduced using simple circuits in one dimension, the SSH transmission line and a braided ladder analogous to the ice-tray model also known as the π-flux ladder. Then, more general building blocks (loops and stars) and wiring schemes (m-shifts) are introduced. The key concepts of emergent pseudo-spin degrees of freedom and synthetic gauge fields are discussed, and the connection to quantum lattice Hamiltonians is clarified. A diagrammatic notation is introduced to simplify the design and presentation of more complicated circuits. These building blocks are then used to construct topological circuits in higher dimensions. The examples include the circuit analog of Haldane’s Chern insulator in two dimensions and quantum Hall insulator in four dimensions featuring finite second Chern numbers. The topological invariants and symmetry protection of the edge modes are discussed based on the H-matrix.
Cosmologies based on General Relativity encompassing an anti-symmetric connection (torsion) can display nice desirable features as the absence of the initial singularity and the possibility of inflation in the early stage of the universe. After briefly reviewing the standard approach to the cosmology with torsion, we generalize it to demonstrate that several theories of torsion gravity are possible using different choices of the diffeomorphic invariants in the Lagrangians. As a result, distinct cosmologies emerge. In all of them it is possible that the universe avoids the initial singularity and passes through an initial accelerated expansion. Differences between these theories are highlighted.
In modern physics there exist several formulations of relativistic thermodynamics of a moving body. The Planck and Ott formalisms are the main ones. However, it is not clear which one is correct. In the present paper, we have solved this problem. We have required the equivalence of the dynamical Hamiltonian of a system to the fundamental thermodynamic potential in addition to the principle of entropy invariance and derived the first law of thermodynamics from this fundamental potential. We have found that in the case of momentum being an independent variable in the Hamiltonian, the Lorentz transformations of the thermodynamic quantities belong to the Planck formalism. However, if we suppose that the velocity is an independent variable in the Hamiltonian (though it is not correct from the point of view of the relativistic dynamics), the Lorentz transformations of the thermodynamic quantities belong to the Ott formalism. It demonstrates that the Ott formalism cannot be appropriate. Moreover, we have proved that in the Planck description the first law of thermodynamics is covariant and the Legendre transform of the Lagrangian is preserved. However, in the Ott description the first law of thermodynamics is not covariant and the Legendre transform is violated. Thus we have demonstrated that only the Planck formulation of relativistic thermodynamics of a moving body is properly defined and the Ott formalism should be discarded.
In this work we develop a mathematical formalism which allows obtaining oscillation formula for neutrino of any energy. We demonstrate that in the ultra-relativistic limit the results obtained in this new approach agree with the previously used phenomenological theory which is only applicable to ultra-relativistic neutrinos. To this end we do the following. The Hilbert spaces of particle states are constructed in such a way that the neutrinos are combined in a multiplet with its components being considered as different quantum states of a single particle. The same is done for the charged leptons and the down- and up-type quarks. In the theory based on the Lagrangian of the fermion sector of the Standard Model modified in accordance with this construction, the phenomenon of neutrino oscillations arises as a direct consequence of the general principles of quantum field theory. Using the example of the pion decay, when the resulting neutrino has to be ultra-relativistic, it is shown that the neutrino states produced in the decay process can be described by a superposition of states with different masses and identical canonical momenta with very high accuracy.
Stabilizer codes are a powerful method for implementing fault-tolerant quantum memory and in the case of topological codes, they form useful models for topological phases of matter. In this paper, we discuss the theory of stabilizer codes as a discrete version of a linear gauge structure, a concept we introduce here. A linear gauge structure captures all the familiar ingredients of U(1) gauge theory including a generalization of charge conservation, Maxwell equations and topological terms. Using this connection, we prove some important results for stabilizer codes which can be used to characterize their error-correction properties. However, this perspective does not depend on any particular Hamiltonian or Lagrangian, which as a consequence is agnostic to the source of the gauge redundancy. Based upon the connection to stabilizer codes, we view the source of the gauge redundancy as an ambiguity in the tensor product structure of the Hilbert space. That is, the gauge provides a set of equivalent but arbitrary ways to factorize the Hilbert space. We apply this formalism to the d=2 and 3 toric code as well as the paradigm fracton models, X-Cube and Haah’s cubic code. From this perspective, we are able to map all these models to some continuum version which captures all or some of the fractonic and topological features of their parent discrete models.