scholarly journals Normal modes of a waveguide as eigenvectors of a self-adjoint operator pencil

2021 ◽  
Vol 29 (1) ◽  
pp. 14-21
Author(s):  
Mikhail D. Malykh
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Wave Equations ◽  
Adjoint Operator ◽  
Normal Modes ◽  
Operator Pencil ◽  
Homogeneous Case ◽  
Permittivity And Permeability ◽  
Simply Connected ◽  
Magnetic Potentials

A waveguide with a constant, simply connected section S is considered under the condition that the substance filling the waveguide is characterized by permittivity and permeability that vary smoothly over the section S, but are constant along the waveguide axis. Ideal conductivity conditions are assumed on the walls of the waveguide. On the basis of the previously found representation of the electromagnetic field in such a waveguide using 4 scalar functions, namely, two electric and two magnetic potentials, Maxwells equations are rewritten with respect to the potentials and longitudinal components of the field. It appears possible to exclude potentials from this system and arrive at a pair of integro-differential equations for longitudinal components alone that split into two uncoupled wave equations in the optically homogeneous case. In an optically inhomogeneous case, this approach reduces the problem of finding the normal modes of a waveguide to studying the spectrum of a quadratic self-adjoint operator pencil.

The twin properties of rogue waves and homoclinic solutions for some nonlinear wave equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wei Tan ◽  
Zhao-Yang Yin
Keyword(s):  
Differential Equations ◽  
Wave Equations ◽  
Rogue Wave ◽  
Nonlinear Partial Differential Equations ◽  
Rogue Waves ◽  
Homoclinic Solutions ◽  
Nonlinear Wave Equations ◽  
Bilinear Method ◽  
Wave Solutions ◽  
Rogue Wave Solutions

Abstract The parameter limit method on the basis of Hirota’s bilinear method is proposed to construct the rogue wave solutions for nonlinear partial differential equations (NLPDEs). Some real and complex differential equations are used as concrete examples to illustrate the effectiveness and correctness of the described method. The rogue waves and homoclinic solutions of different structures are obtained and simulated by three-dimensional graphics, respectively. More importantly, we find that rogue wave solutions and homoclinic solutions appear in pairs. That is to say, for some NLPDEs, if there is a homoclinic solution, then there must be a rogue wave solution. The twin phenomenon of rogue wave solutions and homoclinic solutions of a class of NLPDEs is discussed.

scholarly journals Some arguments for the wave equation in Quantum theory

2021 ◽  
Vol 5 (1) ◽  
pp. 314-336
Author(s):  
Tristram de Piro ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Quantum Theory ◽  
Continuity Equation ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Atomic System ◽  
Balmer Series ◽  
Inertial Frames ◽  
The Stability

We clarify some arguments concerning Jefimenko’s equations, as a way of constructing solutions to Maxwell’s equations, for charge and current satisfying the continuity equation. We then isolate a condition on non-radiation in all inertial frames, which is intuitively reasonable for the stability of an atomic system, and prove that the condition is equivalent to the charge and current satisfying certain relations, including the wave equations. Finally, we prove that with these relations, the energy in the electromagnetic field is quantised and displays the properties of the Balmer series.

An Synergistic Dynamic 2D FEM Model of an Active Magnetic Bearing with Three Electromagnets

2014 ◽  
Vol 214 ◽  
pp. 106-112 ◽  
Author(s):  
Adam Krzysztof Pilat
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Electromagnetic Force ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Comsol Multiphysics ◽  
Mesh Deformation ◽  
Arbitrary Lagrangian Eulerian ◽  
Fem Model ◽  
Mathematical Formulas

This elaboration presents a dynamic model of an Active Magnetic Bearing (AMB) developed in COMSOL Multiphysics. The electromagnetic field is calculated on the basis of Partial Differential Equations (PDEs). The calculated electromagnetic force is applied to the rotor, which is free to move. The Arbitrary Lagrangian-Eulerian (ALE) method for mesh deformation is applied to achieve rotor motion on the bearing plane. The planar rotor motion is described by a set of Ordinary Differential Equations (ODEs) solved in parallel to the electromagnetic field calculations. To enable rotor levitation, three local PD controllers are applied. The mathematical formulas of the control action are coded in the form of COMSOL equations and embedded into the rotor motion ODEs.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

scholarly journals Some applications of the Riesz potential to the theory of the electromagnetic field and the meson field

1946 ◽  
Vol 188 (1012) ◽  
pp. 18-31 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Wave Equation ◽  
Differential Equations ◽  
Riesz Potential ◽  
Analytical Continuation ◽  
Neutral Meson ◽  
Meson Field ◽  
Hyperbolic Differential Equations ◽  
Proper Energy

This paper contains some applications of the method of Marcel Riesz in the solution of normal hyperbolic differential equations, in particular the wave equation, where the known difficulties, due to the occurrence of divergent integrals, are avoided by a process of analytical continuation. In the theory of the electromagnetic field the method yields simple deductions of classical results, but also the results recently obtained by Dirac regarding the proper energy and proper momentum of an electron are obtained without any addition of new assumptions. The corresponding problem in Bhabha’s analogous theory for the neutral meson field are also studied.

scholarly journals Charged radiating stars with Lie symmetries

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
G. Z. Abebe ◽  
S. D. Maharaj
Keyword(s):  
Electromagnetic Field ◽  
Differential Equations ◽  
Equations Of State ◽  
Lie Symmetries ◽  
Boundary Equation ◽  
Symmetries Of Differential Equations ◽  
Radiating Stars ◽  
Barotropic Equation ◽  
Radiating Star ◽  
General Relativistic

Abstract We consider the general model of an accelerating, expanding and shearing radiating star in the presence of charge. Using a new set of variables arising from the Lie symmetries of differential equations we transform the boundary equation into ordinary differential equations. We present several new exact models for a charged gravitating sphere. A particular family of solution may be interpreted as a generalised Euclidean star in the presence of the electromagnetic field. This family admits a linear barotropic equation of state. In the uncharged limit, we regain general relativistic stellar models where proper and areal radii are equal, and its generalisations. Our group theoretical approach selects the physically important cases of Euclidean stars and equations of state.

scholarly journals Stability on the basis of Orthogonal Trajectories

1965 ◽  
Vol 8 (5) ◽  
pp. 647-658
Author(s):  
T. A. Burton
Keyword(s):  
Singular Point ◽  
Differential Equations ◽  
Second Order ◽  
System Of Differential Equations ◽  
Partial Derivatives ◽  
Connected Set ◽  
Image Position ◽  
Simply Connected

We consider a system of differential equations of second order given by1(' = d/dt) where P and Q have continuous first partial derivatives with respect to x and y in some open and simply connected set R containing O = (0, 0) which we assume to be the only singular point in R. In fact, let R be the whole plane; for if not then the following discussion can be modified.

scholarly journals On a class of linear ordinary differential equations having∑k=0∞xkδ(k)(t)and∑k=0mxkδ(k)(t)as solutions

1996 ◽  
Vol 19 (3) ◽  
pp. 555-562
Author(s):  
Ahmed D. Alawneh ◽  
Ahmed Y. Farah
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Rational Solution ◽  
Rational Solutions ◽  
Homogeneous Case ◽  
Linear Ordinary Differential Equations ◽  
Partial Sum ◽  
Distributional Solutions ◽  
Finite Solution

We introduced some linear homogeneous ordinary differential equations which have both formal and finite distributional solutions at the same time, where the finite solution is a partial sum of the formal one. In the nonhomogeneous case and sometimes in the homogeneous case we found formal rational and rational solutions for such differential equations and similarly the rational solution is a partial sum of the formal one.

scholarly journals New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yusuf Pandir ◽  
Halime Ulusoy
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Wave Equations ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equations ◽  
Hyperbolic Function ◽  
Hyperbolic Functions ◽  
Partial Differential ◽  
New Exact Solutions

We firstly give some new functions called generalized hyperbolic functions. By the using of the generalized hyperbolic functions, new kinds of transformations are defined to discover the exact approximate solutions of nonlinear partial differential equations. Based on the generalized hyperbolic function transformation of the generalized KdV equation and the coupled equal width wave equations (CEWE), we find new exact solutions of two equations and analyze the properties of them by taking different parameter values of the generalized hyperbolic functions. We think that these solutions are very important to explain some physical phenomena.

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