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On the existence of non-polarized azimuthal-symmetric electromagnetic waves in circular dielectric waveguide filled with nonlinear isotropic homogeneous medium

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WAVE MOTION 2018年 77卷 77-90页

作者： Smirnov, Yury Smolkin, Eugene Penza State Univ Dept Math & Supercomp Krasnaya Str 40 Penza 440026 Russia

The propagation of monochromatic nonlinear symmetric hybrid waves in a cylindrical nonlinear dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The problem is reduced to the new type of nonlinear eigenvalue problem. The analytical method of solving this problem is presented. New propagation regime is found. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Non-polarized azimuthal-symmetric electromagnetic waves Kerr nonlinearity Non-linear waveguide Maxwell's equations two-parameter eigenvalue problem

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Minimal determinantal representations of bivariate polynomials

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LINEAR ALGEBRA AND ITS APPLICATIONS 2017年 532卷 550-569页

作者： Plestenjak, Bor Univ Ljubljana IMFM Jadranska 19 SI-1000 Ljubljana Slovenia Univ Ljubljana Fac Math & Phys Jadranska 19 SI-1000 Ljubljana Slovenia

For a square-free bivariate polynomial p of degree n we introduce a simple and fast numerical algorithm for the construction of n x n matrices A, B, and C such that det(A+xB+yC) = p(x, y). This is the minimal size needed to represent a bivariate polynomial of degree n. Combined with a square-free factorization one can now compute n x n matrices for any bivariate polynomial of degree n. The existence of such symmetric matrices was established by Dixon in 1902, but, up to now, no simple numerical construction has been found, even if the matrices can be nonsymmetric. Such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two-parameter eigenvalue problem. The new representation speeds up the computation considerably. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Bivariate polynomial Determinantal representation System of bivariate polynomial equations two-parameter eigenvalue problem

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SOLVING GENERALIZED CDT problemS VIA two-parameter eigenvalueS

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SIAM JOURNAL ON OPTIMIZATION 2016年第3期26卷 1669-1694页

作者： Sakaue, Shinsaku Nakatsukasa, Yuji Takeda, Akiko Iwata, Satoru Univ Tokyo Dept Math Informat Tokyo 1138656 Japan

We consider solving a nonconvex quadratic minimization problem with two quadratic constraints, one of which being convex. This problem is a generalization of the Celis-Denis-Tapia (CDT) problem and thus we refer to it as GCDT (Generalized CDT). The CDT problem has been widely studied, but no polynomial-time algorithm was known until Bienstock's recent work. His algorithm solves the CDT problem in polynomial time with respect to the number of bits in data and log is an element of(-1) by admitting an is an element of error in the constraints. The algorithm, however, appears to be difficult to implement. In this paper, we present another algorithm for GCDT, which is guaranteed to find a global solution for almost all GCDT instances (and slightly perturbed ones in some exceptionally rare cases), in exact arithmetic (including eigenvalue computation). Our algorithm is based on the approach proposed by Iwata, Nakatsukasa, and Takeda (2015) for computing the signed distance between overlapping ellipsoids. Our algorithm computes all the Lagrange multipliers of GCDT by solving a two-parameter linear eigenvalue problem, obtains the corresponding KKT points, and finds a global solution as the KKT point with the smallest objective value. In practice, in finite precision arithmetic, our algorithm requires O(n(6) log log u(-1)) computational time, where n is the number of variables and u is the unit roundoff. Although we derive our algorithm under the unrealistic assumption that exact eigenvalues can be computed, numerical experiments illustrate that our algorithm performs well in finite precision arithmetic.

关键词： quadratically constrained quadratic programming nonconvex optimization Celis-Dennis-Tapia problem two-parameter eigenvalue problem

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On the Existence of Nonlinearizable Solutions of a Nonclassicaltwo-parameter Nonlinear Boundary Value problem

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DIFFERENTIAL EQUATIONS 2024年第4期60卷 445-464页

作者： Martynova, V. Yu. Penza State Univ Penza 440026 Russia

A nonlinear eigenvalue problem for a system of three equations with boundary conditions of the first kind, describing the propagation of electromagnetic waves in a plane nonlinear waveguide, is considered. This is a two-parameter problem with one spectral parameter and a second parameter arising from an additional condition. This condition connects the integration constants that arise when finding the first integrals of the system. The existence of nonlinearizable solutions of the problem is proved.

关键词： two-parameter eigenvalue problem nonlinearizable solution integral characteristic equation Maxwell equations nonlinear permittivity

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Perturbation Method in the Theory of Propagation of two-Frequency Electromagnetic Waves in a Nonlinear Waveguide I: TE-TE Waves

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2021年第1期61卷 103-117页

作者： Valovik, D. V. Penza State Univ Penza 440026 Russia

The article deals with the problem of propagation of a two-frequency electromagnetic wave in a waveguide filled with a nonlinear medium. A two-frequency wave is the sum of two monochromatic TE waves with different frequencies. The permittivity of the waveguide is characterized by a very general nonlinearity function corresponding to self-action effects. It is shown that, under certain conditions, the two-frequency wave is an eigenmode of the waveguide. From a mathematical point of view, the problem reduces to a nonlinear two-parameter eigenvalue problem for the system of (nonlinear) Maxwell's equations. The main result of the article is the proof of the existence of nonlinearizable solutions of the problem.

关键词： Maxwell’ s equations nonlinear permittivity planar dielectric waveguide nonlinear Sturm– Liouville-type problem two-parameter eigenvalue problem perturbation method integral characteristic equation

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HARMONIC RAYLEIGH-RITZ EXTRACTION FOR THE MULTIparameter eigenvalue problem

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2007年 29卷 81-96页

作者： Hochstenbach, Michiel E. Plestenjak, Bor Eindhoven Univ Technol Dept Math & Comp Sci NL-5600 MB Eindhoven Netherlands Univ Ljubljana Dept Math SI-1000 Ljubljana Slovenia

We study harmonic and refined extraction methods for the multiparameter eigenvalue problem. These techniques are generalizations of their counterparts for the standard and generalized eigenvalue problem. The methods aim to approximate interior eigenpairs, generally more accurately than the standard extraction does. We study their properties and give Saad-type theorems. The processes can be combined with any subspace expansion approach, for instance a Jacobi-Davidson type technique, to form a subspace method for multiparameter eigenproblems of high dimension.

关键词： multiparameter eigenvalue problem two-parameter eigenvalue problem harmonic extraction refined extraction Rayleigh-Ritz subspace method Saad's theorem Jacobi-Davidson

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Nonclassical perturbation approach to a nonlinear multiparameter eigenvalue problem arising in electromagnetics

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第12期46卷 13205-13225页

作者： Martynova, Valeria Valovik, Dmitry Penza State Univ Dept Math & Supercomp Krasnaya ul 40 Penza 440026 Russia

The paper focuses on a problem of nonlinear electromagnetic wave propagation in a plane shielded waveguide filled with nonlinear medium. We look for nonlinear eigenvalues that correspond to propagation constants of the waveguide. One of the main feature of the problem is that it has nonperturbative solutions and therefore the corresponding linearized problem cannot be used in any perturbation approach to find such eigenvalues (moreover, it turns out that the linearized problem is solvable in exceptional cases only). To solve the main problem, we develop a nonlinear perturbation approach based on usage of a certain simpler nonlinear problem that also has nonlinearizable solutions. Using this approach, we prove existence of nonlinearizable solutions to the main problem. Numerical results are also presented. Briefly, we present electromagnetic background for the main problem.

关键词： integral characteristic equation Maxwell's equations nonlinear eigenvalue problem nonlinear permittivity nonpertubative solutions ordinary differential equations two-parameter eigenvalue problem

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A Jacobi-Davidson method for two-real-parameter nonlinear eigenvalue problems arising from delay-differential equations

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NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS 2013年第5期20卷 852-868页

作者： Meerbergen, Karl Schroeder, Christian Voss, Heinrich Katholieke Univ Leuven Dept Comp Sci B-3001 Heverlee Belgium TU Berlin Inst Math Berlin Germany Hamburg Univ Technol Inst Numer Simulat Hamburg Germany

The critical delays of a delay-differential equation can be computed by solving a nonlinear two-parameter eigenvalue problem. The solution of this two-parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR-type method for solving such quadratic eigenvalue problem that only computes real-valued critical delays;that is, complex critical delays, which have no physical meaning, are discarded. For large-scale problems, we propose new correction equations for a Newton-type or Jacobi-Davidson style method, which also forces real-valued critical delays. We present three different equations: one real-valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi-Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large-scale problems arising from PDEs. Copyright (C) 2012 John Wiley & Sons, Ltd.

关键词： two-parameter eigenvalue problem Jacobi-Davidson nonlinear eigenvalue problem delay-differential equation critical delay

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About one gradient procedure of determination of the branching points of nonlinear integral equation arising in the theory of antenna synthesis 4

About one gradient procedure of determination of the branchi...

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4th International Conference on Antenna Theory and Techniques

作者： Podlevskyi, BM NASU Pidstryhach Inst Appl Problems Mech & Math UA-79000 Lvov Ukraine

ISBN: (纸本)0780378814

The problem of building a numerical algorithm for determination of branching points of one nonlinear integral operator, which arises in the theory of antennas synthesis according to the given amplitude radiation pattern, is considered. The basic difficulty, consists in that the kernel of an integral operator nonlinearly depends oil two parameters, which play role of the spectral ones. For such problems, except a special case. the existing numerical algorithms are not applicable. For building an algorithm for solving such problems, the equivalent variational statement is used. The problem is reduced to a sequence of linear two-parameter eigenvalue problems with application of one gradient procedure for simultaneous evaluation of two spectral parameters being the branching points of an initial nonlinear integral operator.

关键词： antennas synthesis branching points nonlinear integral operator two-parameter eigenvalue problem numerical algorithm eigenvalue gradient procedure

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Electromagnetic Non-Polarized Symmetric Hybrid Wave Propagation in a Plane Waveguide with Nonlinear Anisotropic Permittivity

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LOBACHEVSKII JOURNAL OF MATHEMATICS 2018年第8期39卷 1075-1089页

作者： Kurseeva, V. Yu. Penza State Univ Krasnaya Ul 40 Penza 440026 Russia

Paper focuses on the propagation of monochromatic nonlinear symmetric hybrid waves in a planar dielectric waveguide filled with nonlinear anisotropic 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 a perturbation method, it is theoretically proved the existence of a finite number of isolated eigenvalues and, therefore, guide waves. Numerical results are presented.

关键词： Non-polarized symmetric electromagnetic waves Kerr nonlinearity saturated nonlinearity plane nonlinear anisotropic waveguide Maxwell's equations two-parameter eigenvalue problem

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