On the problem of propagation of nonlinear coupled TE-TM waves in a layer

The problem of simultaneous propagation of two types of electromagnetic waves (TE and TM) in a plane dielectric waveguide filled with a nonlinear medium is considered. These polarized waves have different frequencies and different propagation constants. The physical problem is reduced to a nonlinear two-parameter transmission eigenvalue problem for Maxwell’s equations in a layer. The coupled eigenvalues are coupled propagation constants. A theorem on the existence and localization of coupled eigenvalues corresponding to coupled polarized electromagnetic waves is proved.     

The new type of non-polarized symmetric electromagnetic waves in planar nonlinear waveguide

Paper focuses on the propagation of monochromatic nonlinear symmetric hybrid waves in a planar dielectric waveguide filled with nonlinear medium. The wave propagation problem is reduced to a transmission eigenvalue problem. Eigenvalues of the problem depend on an additional parameter and correspond to propagation constant. Using perturbation method, it is theoretically proved the existence of a finite number of isolated eigenvalues and therefore, guide waves. The found guide regime is novel in the theory of nonlinear waveguides. Numerical results are presented.     

Integral dispersion equation method to solve a nonlinear boundary eigenvalue problem

In this work a nonlinear eigenvalue problem for a nonlinear autonomous ordinary differential equation of the second order is considered. This problem describes the process of propagation of transverse-electric electromagnetic waves along a plane dielectric waveguide with nonlinear permittivity. We demonstrate, as far as we know, a new method that allows one to derive an equation w.r.t. spectral parameter (the dispersion equation) which contains all necessary information about the eigenvalues. The method is based on a simple idea that the distance between zeros of a periodic solution to the differential equation is the same for the adjacent zeros. This method has no connections with the perturbation theory or the notion of a bifurcation point. Theorem of equivalence between the eigenvalue problem and the dispersion equation is proved. Periodicity of the eigenfunctions is proved, a formula for the period is found, and zeros of the eigenfunctions are determined. The formula for the distance between adjacent zeros of any eigenfunction is given. Also theorems of existence and localization of the eigenvalues are proved.     

On the existence of infinitely many eigenvalues in a nonlinear Sturm–Liouville problem arising in the theory of waveguides

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type on an interval with boundary conditions of the first kind. The problem describes the propagation of polarized electromagnetic waves in a plane two-layer dielectric waveguide. The cases of a homogeneous and an inhomogeneous medium are studied. The existence of infinitely many positive and negative eigenvalues is proved.     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Nonlinear propagation of coupled electromagnetic waves in a circular cylindrical waveguide

The problem of the propagation of coupled surface electromagnetic waves in a two-layer cylindrical circular waveguide filled with an inhomogeneous nonlinear medium is considered. A nonlinear coupled TE-TM wave is characterized by two (independent) frequencies ω_e and ω_m and two propagation constants \({\widehat \gamma _e}\) and \({\widehat \gamma _m}\). The physical problem reduces to a nonlinear two-parameter eigenvalue problem for a system of nonlinear ordinary differential equations. The existence of eigenvalues (\({\widehat \gamma _e}\), \({\widehat \gamma _m}\)) in proven and intervals of their localization are determined.     

On the existence of infinitely many nonperturbative solutions in a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity

The paper focuses on a transmission eigenvalue problem for nonlinear Helmholtz equation with polynomial nonlinearity which describes the propagation of transverse electric waves along a dielectric layer filled with nonlinear medium. It is proved that even if the nonlinearity coefficients are small, the nonlinear problem has infinitely many nonperturbative solutions, whereas the corresponding linear problem always has a finite number of solutions. This results in the theoretical existence of a novel type of nonlinear guided waves that exist only in nonlinear guided systems. Asymptotic distribution of the eigenvalues is found and a comparison theorem is proved; periodicity of the eigenfunctions is proved, the exact formula for the period is found, and the zeros of the eigenfunctions are determined. The results found essentially extend the theory evolved earlier (particular cases for Kerr, cubic-quintic, septic nonlinearities, etc. are easily extracted from the general results found here). Numerical results are also presented.     

Substantiation of the Spectral Method for Calculating Propagation Constants for Waveguides with Microstrip and Slit Lines

We prove the spectral method for calculating the propagation constants of normal waves in screened waveguides with an irregular cross-section and a nonhomogeneous filling. The problem is reduced to the nonlinear spectral problem for operator-functions in a Hilbert space. Existence theorems are proved for spectral points and the cutoff method is substantiated.     

An initial boundary-value problem for model electromagnetoelasticity system

In this paper we consider an initial boundary-value problem related to the electrodynamics of vibrating elastic media. The aim is to prove an existence and uniqueness result for a model describing the nonlinear interactions of the electromagnetic and elastic waves. We assume that the motion of the continuum occurs at velocities that are much smaller than the propagation velocity of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations, one of them is the hyperbolic equation (an analog of the Lamé system) and another one is the parabolic equation (an analog of the diffusion Maxwell system). One stability result is proved too.     

On the Analytic Solutions of a Special Boundary Value Problem for a Nonlinear Heat Equation in Polar Coordinates

The paper addresses a nonlinear heat equation (the porous medium equation) in the case of a power-law dependence of the heat conductivity coefficient on temperature. The equation is used for describing high-temperature processes, filtration of gases and fluids, groundwater infiltration, migration of biological populations, etc. The heat waves (waves of filtration) with a finite velocity of propagation over a cold background form an important class of solutions to the equation under consideration. A special boundary value problem having solutions of such type is studied. The boundary condition of the problem is given on a sufficiently smooth closed curve with variable geometry. The new theorem of existence and uniqueness of the analytic solution is proved.     

Optimal control of nonlinear one-dimensional periodic wave equation with x-dependent coefficients

This paper is concerned with an optimal control problem governed by the nonlinear one dimensional periodic wave equation with x-dependent coefficients. The control of the system is realized via the outer function of the state. Such a model arises from the propagation of seismic waves in a nonisotropic medium. By investigating some important properties of the linear operator associated with the state equation, we obtain the existence and regularity of the weak solution to the state equation. Furthermore, the existence of the optimal control is proved by means of the Arzelà-Ascoli lemma and Sobolev compact imbedding theorem.     

On the eigenvalues of a nonlinear spectral problem

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type with conditions of the third kind, which describes the propagation of polarized electromagnetic waves in a plane dielectric waveguide. The equation is nonlinear in the unknown function, and the boundary conditions depend on the spectral parameter nonlinearly. We obtain an equation for the spectral parameter and formulas for the zeros of the eigenfunctions and show that the problem has at most finitely many isolated eigenvalues.     

Rotating waves in parabolic functional differential equations with rotation of spatial argument and time delay

The parabolic functional differential equation $\frac{{\partial u}} {{\partial t}} = D\frac{{\partial ^2 u}} {{\partial x^2 }} - u + K(1 + \gamma \cos u(x + \theta ,t - T)) $ is considered on the circle [0, 2π]. Here, D > 0, T > 0, K > 0, and γ ∈ (0, 1). Such equations arise in the modeling of nonlinear optical systems with a time delay T > 0 and a spatial argument rotated by an angle θ ∈ [0, 2π) in the nonlocal feedback loop in the approximation of a thin circular layer. The goal of this study is to describe spatially inhomogeneous rotating-wave solutions bifurcating from a homogeneous stationary solution in the case of a Andronov-Hopf bifurcation. The existence of such waves is proved by passing to a moving coordinate system, which makes it possible to reduce the problem to the construction of a nontrivial solution to a periodic boundary value problem for a stationary delay differential equation. The existence of rotating waves in an annulus resulting from a Andronov-Hopf bifurcation is proved, and the leading coefficients in the expansion of the solution in powers of a small parameter are obtained. The conditions for the stability of waves are derived by constructing a normal form for the Andronov-Hopf bifurcation for the functional differential equation under study.     

On the existence and uniqueness of solutions of the inverse boundary value problem for determining the permittivity of materials

The problem of determining the permittivity of material samples of arbitrary shape placed in a rectangular waveguide with perfectly conducting walls is investigated. The problem is reduced to solving a nonlinear volume singular integral equation. A theorem on the existence and uniqueness of solutions to the nonlinear volume singular integral equation and of the inverse boundary value problem for determining the permittivity of the material is proved.     

The first initial–boundary-value problem for a nonlinear model of the electrodynamics of vibrating elastic media

The first initial–boundary-value problem for nonlinear differential equations describing the interactions of a vibrating electroconductive body and the electromagnetic field is studied. We assume that the motion of the body occurs at velocities that are much smaller than the velocity of propagation of the electromagnetic waves through the elastic medium. The model under study consists of two coupled differential equations; one of them is the hyperbolic equation (an analogue of the Lamé system) and the other is the parabolic equation (an analogue of the diffusion Maxwell system). We prove an existence and uniqueness result. The proof is based on the classical Faedo–Galerkin method.     

Existence and multiplicity of wave trains in 2D lattices

We study the existence and branching patterns of wave trains in a two-dimensional lattice with linear and nonlinear coupling between nearest particles and a nonlinear substrate potential. The wave train equation of the corresponding discrete nonlinear equation is formulated as an advanced-delay differential equation which is reduced by a Lyapunov–Schmidt reduction to a finite-dimensional bifurcation equation with certain symmetries and an inherited Hamiltonian structure. By means of invariant theory and singularity theory, we obtain the small amplitude solutions in the Hamiltonian system near equilibria in non-resonance and p:q$p:q$ resonance, respectively. We show the impact of the direction θ of propagation and obtain the existence and branching patterns of wave trains in a one-dimensional lattice by investigating the existence of traveling waves of the original two-dimensional lattice in the direction θ of propagation satisfying tan θ is rational.     

A radiation condition for uniqueness in a wave propagation problem for 2‐D open waveguides

We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. We provide an explicit condition for uniqueness for rectilinear waveguides, which takes into account the physically significant components, corresponding to guided and non‐guided waves; this condition reduces to the classical Sommerfeld–Rellich condition in the relevant cases. By a careful asymptotic analysis we prove that the solution derived by Magnanini and Santosa (SIAM J. Appl. Math. 2001; 61 :1237–1252) for stratified media satisfies our radiation condition. Copyright © 2008 John Wiley & Sons, Ltd.     

Propagation of weakly guided waves in a Kerr nonlinear medium using a perturbation approach

The equations are represented in a simplified format with only a few leading terms needed in the expansion. The set of equations are then solved numerically using vector finite element method. To validate our algorithm, we analyzed a two-dimensional rectangular waveguide consisting of a linear core and nonlinear identical cladding. The exact nonlinear solutions for three different modes of propagations, TE0, TE1, and TE2 modes are generated and compared with the computed solutions. Next, we investigate the effect of a more intense monochromatic field on the propagation of a “weak” optical field in a fully three-dimensional cylindrical waveguide.     

Existence of a solution of an initial-boundary value difference problem for a linear heat equation with a nonlinear boundary condition

A problem of heat propagation in the ground from a heated pipeline with a partially heat-insulating shell is considered. The possibility is proved to construct a numerical solution of a linear heat equation by using a direct finite-difference method in the case when the thermal radiation on the ground surface is taken into account. On the basis of the theorem about the solvability of a system of linear difference equations by means of the sweep method, the existence and uniqueness of a solution of a corresponding difference problem with nonlinear boundary condition are proved.     

On a class of essentially nonlinear elliptic differential-difference equations

An essentially nonlinear differential-difference equation containing the product of the p-Laplacian and a difference operator is considered. Sufficient conditions are obtained for the corresponding nonlinear differential-difference operator to be coercive and pseudomonotone in the case of nonvariational statement of the differential equation. The existence of a generalized solution to the Dirichlet problem for the nonlinear equation is proved.