Zeitschrift für angewandte Mathematik und Physik, ISSN 0044-2275, 12/2017, Volume 68, Issue 6, pp. 1 - 19

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic...

Standing waves | Engineering | 35J20 | Mathematical Methods in Physics | Nonlinear Helmholtz equations | Oscillating solutions | 35Q55 | 35J05 | Theoretical and Applied Mechanics | EXISTENCE | RADIAL SOLUTIONS | MATHEMATICS, APPLIED | SCALAR FIELD-EQUATIONS | BOUNDED SOLUTIONS | Mathematics - Analysis of PDEs

Standing waves | Engineering | 35J20 | Mathematical Methods in Physics | Nonlinear Helmholtz equations | Oscillating solutions | 35Q55 | 35J05 | Theoretical and Applied Mechanics | EXISTENCE | RADIAL SOLUTIONS | MATHEMATICS, APPLIED | SCALAR FIELD-EQUATIONS | BOUNDED SOLUTIONS | Mathematics - Analysis of PDEs

Journal Article

Journal of Computational Physics, ISSN 0021-9991, 08/2017, Volume 343, p. 1

A new iterative method is developed for solving the two-dimensional nonlinear Helmholtz equation which governs polarized light in media with the optical Kerr...

Helmholtz equations | Broken symmetry | Propagation | Boundary conditions | Optical bistability | Circular cylinders | Polarized light | Spectral methods | Bistability | Robustness (mathematics) | Nonlinearity | Iterative methods | Nonlinear systems | Cylinders

Helmholtz equations | Broken symmetry | Propagation | Boundary conditions | Optical bistability | Circular cylinders | Polarized light | Spectral methods | Bistability | Robustness (mathematics) | Nonlinearity | Iterative methods | Nonlinear systems | Cylinders

Journal Article

Computer Methods in Applied Mechanics and Engineering, ISSN 0045-7825, 04/2019, Volume 346, pp. 260 - 287

We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed...

Helmholtz equation | A posteriori error estimate | Adaptive algorithm | Optimality | Boundary element method | Convergence | ALGORITHM | INTEGRAL-EQUATIONS | FEM | POSTERIORI ERROR ESTIMATE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | BOUNDARY-ELEMENT METHODS | EFFICIENCY | SCATTERING | SURFACES | Operators (mathematics) | Helmholtz equations | Mathematical analysis | Integral equations | Adaptive algorithms | Nonlinear programming | Mathematics - Numerical Analysis

Helmholtz equation | A posteriori error estimate | Adaptive algorithm | Optimality | Boundary element method | Convergence | ALGORITHM | INTEGRAL-EQUATIONS | FEM | POSTERIORI ERROR ESTIMATE | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | BOUNDARY-ELEMENT METHODS | EFFICIENCY | SCATTERING | SURFACES | Operators (mathematics) | Helmholtz equations | Mathematical analysis | Integral equations | Adaptive algorithms | Nonlinear programming | Mathematics - Numerical Analysis

Journal Article

Computers and Mathematics with Applications, ISSN 0898-1221, 2011, Volume 62, Issue 10, pp. 3894 - 3901

Approximate periodic solutions for the Helmholtz–Duffing oscillator are obtained in this paper. He’s Energy Balance Method (HEBM) and He’s Frequency Amplitude...

Helmholtz–Duffing oscillator | He’s Frequency Amplitude Formulation | He’s Energy Balance Method | He's Energy Balance Method | HelmholtzDuffing oscillator | He's Frequency Amplitude Formulation | ENERGY-BALANCE | MATHEMATICS, APPLIED | HOMOTOPY PERTURBATION | VARIATIONAL APPROACH | Helmholtz-Duffing oscillator | VIBRATIONS | NONLINEAR OSCILLATORS | HES FREQUENCY FORMULATION | U(1/3) FORCE | Energy of formation | Approximation | Error analysis | Mathematical analysis | Resonant frequency | Oscillations | Mathematical models | Oscillators

Helmholtz–Duffing oscillator | He’s Frequency Amplitude Formulation | He’s Energy Balance Method | He's Energy Balance Method | HelmholtzDuffing oscillator | He's Frequency Amplitude Formulation | ENERGY-BALANCE | MATHEMATICS, APPLIED | HOMOTOPY PERTURBATION | VARIATIONAL APPROACH | Helmholtz-Duffing oscillator | VIBRATIONS | NONLINEAR OSCILLATORS | HES FREQUENCY FORMULATION | U(1/3) FORCE | Energy of formation | Approximation | Error analysis | Mathematical analysis | Resonant frequency | Oscillations | Mathematical models | Oscillators

Journal Article

Communications in Mathematical Physics, ISSN 0010-3616, 6/2019, Volume 368, Issue 2, pp. 799 - 842

We prove an L p -version of the limiting absorption principle for a class of periodic elliptic differential operators of second order. The result is applied to...

Quantum Physics | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | PHYSICS, MATHEMATICAL

Quantum Physics | Mathematical Physics | Classical and Quantum Gravitation, Relativity Theory | Theoretical, Mathematical and Computational Physics | Complex Systems | Physics | PHYSICS, MATHEMATICAL

Journal Article

Applied Mathematical Modelling, ISSN 0307-904X, 09/2018, Volume 61, pp. 593 - 603

•A new method to obtain approximate fully analytical steady state solutions of NLODEs is presented.•The method is applied to the forced and damped...

Nonlinear dynamics | Fourier series | Harmonic balance method | Helmholtz–Duffing equation | Anharmonic oscillator | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | RESONATORS | SYSTEMS | Helmholtz-Duffing equation | HARMONIC-BALANCE | Differential equations

Nonlinear dynamics | Fourier series | Harmonic balance method | Helmholtz–Duffing equation | Anharmonic oscillator | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | ENGINEERING, MULTIDISCIPLINARY | RESONATORS | SYSTEMS | Helmholtz-Duffing equation | HARMONIC-BALANCE | Differential equations

Journal Article

ESAIM: Mathematical Modelling and Numerical Analysis, ISSN 0764-583X, 05/2016, Volume 50, Issue 3, pp. 783 - 808

We introduce and analyze a virtual element method (VEM) for the Helmholtz problem with approximating spaces made of products of low order VEM functions and...

Duality estimates | Helmholtz equation | Error analysis | Plane wave basis functions | Virtual element method | virtual element method | MATHEMATICS, APPLIED | error analysis | LAGRANGE MULTIPLIERS | TREFFTZ | duality estimates | WEAK VARIATIONAL FORMULATION | ACOUSTICS | BOUNDS | DISCONTINUOUS GALERKIN METHODS | POLYGONAL MESHES | VERSION | plane wave basis functions | LINEAR ELASTICITY PROBLEMS | EQUATION | Helmholtz equations | Basis functions | Apexes | Mathematical analysis | Plane waves | Boundary conditions | Nonlinear programming

Duality estimates | Helmholtz equation | Error analysis | Plane wave basis functions | Virtual element method | virtual element method | MATHEMATICS, APPLIED | error analysis | LAGRANGE MULTIPLIERS | TREFFTZ | duality estimates | WEAK VARIATIONAL FORMULATION | ACOUSTICS | BOUNDS | DISCONTINUOUS GALERKIN METHODS | POLYGONAL MESHES | VERSION | plane wave basis functions | LINEAR ELASTICITY PROBLEMS | EQUATION | Helmholtz equations | Basis functions | Apexes | Mathematical analysis | Plane waves | Boundary conditions | Nonlinear programming

Journal Article

ADVANCED NONLINEAR STUDIES, ISSN 1536-1365, 08/2019, Volume 19, Issue 3, pp. 569 - 593

We obtain uncountably many solutions of nonlinear Helmholtz and curl-curl equations on the entire space using a fixed point approach. The constructed solutions...

MATHEMATICS | MATHEMATICS, APPLIED | Nonlinear Helmholtz Equations | HARMONIC MAXWELL EQUATIONS | Curl-Curl Equations | Herglotz Waves | ABSENCE | Limiting Absorption Principles | OPERATORS | GROUND-STATES

MATHEMATICS | MATHEMATICS, APPLIED | Nonlinear Helmholtz Equations | HARMONIC MAXWELL EQUATIONS | Curl-Curl Equations | Herglotz Waves | ABSENCE | Limiting Absorption Principles | OPERATORS | GROUND-STATES

Journal Article

Nonlinear Analysis, ISSN 0362-546X, 03/2017, Volume 152, pp. 88 - 101

In the first part of this paper, the existence of infinitely many Lp-standing wave solutions for the nonlinear Helmholtz equation −Δu−λu=Q(x)∣u∣p−2u in RN is...

Nonlinear Helmholtz equation | Pseudo-gradient flow | Variational method | SCHRODINGER-EQUATIONS | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | SEMILINEAR ELLIPTIC-EQUATIONS | INDEFINITE LINEAR PART | MULTIBUMP SOLUTIONS

Nonlinear Helmholtz equation | Pseudo-gradient flow | Variational method | SCHRODINGER-EQUATIONS | EXISTENCE | MATHEMATICS | MATHEMATICS, APPLIED | MULTIPLE SOLUTIONS | SEMILINEAR ELLIPTIC-EQUATIONS | INDEFINITE LINEAR PART | MULTIBUMP SOLUTIONS

Journal Article

Advances in Mathematics, ISSN 0001-8708, 08/2015, Volume 280, pp. 690 - 728

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation−Δu−k2u=Q(x)|u|p−2u,u∈W2,p(RN) with N≥3,...

Standing waves | Nonlinear Helmholtz equation | Dual variational method | Nonvanishing | EXISTENCE | MATHEMATICS | SCALAR FIELD-EQUATIONS | CALCULUS | CONCENTRATION-COMPACTNESS PRINCIPLE | OPERATORS | Mathematics - Analysis of PDEs

Standing waves | Nonlinear Helmholtz equation | Dual variational method | Nonvanishing | EXISTENCE | MATHEMATICS | SCALAR FIELD-EQUATIONS | CALCULUS | CONCENTRATION-COMPACTNESS PRINCIPLE | OPERATORS | Mathematics - Analysis of PDEs

Journal Article

Proceedings of the Royal Society of Edinburgh Section A: Mathematics, ISSN 0308-2105, 2019, pp. 1 - 32

Abstract Using a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation $$-\Delta u-k^2u = Q(x)...

nonvanishing | dual variational method | nonexistence results | Nonlinear Helmholtz equation | critical exponent

nonvanishing | dual variational method | nonexistence results | Nonlinear Helmholtz equation | critical exponent

Journal Article

12.
Full Text
Lie symmetry analysis and group invariant solutions of the nonlinear Helmholtz equation

Applied Mathematics and Computation, ISSN 0096-3003, 08/2018, Volume 331, pp. 457 - 472

We consider the nonlinear Helmholtz (NLH) equation describing the beam propagation in a planar waveguide with Kerr-like nonlinearity under non-paraxial...

Symmetry reduction | Periodic and solitary waves | Painlevé analysis | Modified Prelle–Singer method | Nonlinear Helmholtz equation | Lie symmetry analysis | MATHEMATICS, APPLIED | INTEGRABILITY | LINEAR SCHRODINGER-EQUATION | Painleveanalysis | EVOLUTION-EQUATIONS | REDUCTIONS | WAVES | SOLITONS | Modified Prelle-Singer method | SYSTEMS | ORDINARY DIFFERENTIAL-EQUATIONS | PROPAGATION

Symmetry reduction | Periodic and solitary waves | Painlevé analysis | Modified Prelle–Singer method | Nonlinear Helmholtz equation | Lie symmetry analysis | MATHEMATICS, APPLIED | INTEGRABILITY | LINEAR SCHRODINGER-EQUATION | Painleveanalysis | EVOLUTION-EQUATIONS | REDUCTIONS | WAVES | SOLITONS | Modified Prelle-Singer method | SYSTEMS | ORDINARY DIFFERENTIAL-EQUATIONS | PROPAGATION

Journal Article

Applied Mathematics Letters, ISSN 0893-9659, 12/2012, Volume 25, Issue 12, pp. 2349 - 2353

In this paper, we derive a class of analytical solution of the damped Helmholtz–Duffing oscillator that is based on a recently developed exact solution for the...

Damped Helmholtz–Duffing oscillator | Mixed-parity nonlinear oscillator | Asymmetric behavior | Quadratic nonlinearities | Damped Helmholtz-Duffing oscillator | MATHEMATICS, APPLIED | INTEGRABILITY | OSCILLATOR | Mathematical analysis | Oscillators | Exact solutions

Damped Helmholtz–Duffing oscillator | Mixed-parity nonlinear oscillator | Asymmetric behavior | Quadratic nonlinearities | Damped Helmholtz-Duffing oscillator | MATHEMATICS, APPLIED | INTEGRABILITY | OSCILLATOR | Mathematical analysis | Oscillators | Exact solutions

Journal Article

Numerische Mathematik, ISSN 0029-599X, 6/2018, Volume 139, Issue 2, pp. 479 - 503

The paper is concerned with the unconditional stability and optimal $$L^2$$ L2 error estimates of linearized Crank–Nicolson Galerkin FEMs for a nonlinear...

65N15 | Theoretical, Mathematical and Computational Physics | Unconditionally optimal error estimates | Mathematics | Mathematical Methods in Physics | Numerical Analysis | Mathematical and Computational Engineering | Nonlinear Schrödinger–Helmhotz equations | Mathematics, general | 65M60 | Numerical and Computational Physics, Simulation | Linearized Crank–Nicolson Galerkin FEMs | 65N30 | MATHEMATICS, APPLIED | Nonlinear Schrodinger-Helmhotz equations | NUMERICAL-SOLUTION | POISSON | APPROXIMATIONS | Linearized Crank-Nicolson Galerkin FEMs | ERROR ANALYSIS | DIFFERENCE-SCHEMES | FINITE-ELEMENT-METHOD | EQUATION

65N15 | Theoretical, Mathematical and Computational Physics | Unconditionally optimal error estimates | Mathematics | Mathematical Methods in Physics | Numerical Analysis | Mathematical and Computational Engineering | Nonlinear Schrödinger–Helmhotz equations | Mathematics, general | 65M60 | Numerical and Computational Physics, Simulation | Linearized Crank–Nicolson Galerkin FEMs | 65N30 | MATHEMATICS, APPLIED | Nonlinear Schrodinger-Helmhotz equations | NUMERICAL-SOLUTION | POISSON | APPROXIMATIONS | Linearized Crank-Nicolson Galerkin FEMs | ERROR ANALYSIS | DIFFERENCE-SCHEMES | FINITE-ELEMENT-METHOD | EQUATION

Journal Article

International Journal of Nonlinear Sciences and Numerical Simulation, ISSN 1565-1339, 2006, Volume 7, Issue 3, pp. 321 - 328

In this article, He's homotopy perturbation method (HPM), which does not need small parameter in the equation, is implemented to solve the linear Helmholtz...

Nonlinear partial differential equations | Homotopy perturbation method | Heimholtz equation | FKdV equation | WATER-WAVES | MATHEMATICS, APPLIED | MECHANICS | nonlinear partial differential equations | ENGINEERING, MULTIDISCIPLINARY | NONLINEAR PROBLEMS | DECOMPOSITION METHOD | Helmholtz equation | homotopy perturbation method | PHYSICS, MATHEMATICAL | BIFURCATION

Nonlinear partial differential equations | Homotopy perturbation method | Heimholtz equation | FKdV equation | WATER-WAVES | MATHEMATICS, APPLIED | MECHANICS | nonlinear partial differential equations | ENGINEERING, MULTIDISCIPLINARY | NONLINEAR PROBLEMS | DECOMPOSITION METHOD | Helmholtz equation | homotopy perturbation method | PHYSICS, MATHEMATICAL | BIFURCATION

Journal Article

SIAM Journal on Numerical Analysis, ISSN 0036-1429, 2018, Volume 56, Issue 3, pp. 1338 - 1359

The well-posedness of a nonlinear Helmholtz equation with an impedance boundary condition is established for high frequencies in two and three dimensions....

Finite element method | Error estimates | Stability | Nonlinear Helmholtz equation | High wave number | MATHEMATICS, APPLIED | CONVERGENCE ANALYSIS | high wave number | BOUNDARY-CONDITIONS | PREASYMPTOTIC ERROR ANALYSIS | DISCRETIZATIONS | NUMERICAL-SOLUTION | finite element method | PENALTY | nonlinear Helmholtz equation | POLLUTION | CIP-FEM | error estimates | stability

Finite element method | Error estimates | Stability | Nonlinear Helmholtz equation | High wave number | MATHEMATICS, APPLIED | CONVERGENCE ANALYSIS | high wave number | BOUNDARY-CONDITIONS | PREASYMPTOTIC ERROR ANALYSIS | DISCRETIZATIONS | NUMERICAL-SOLUTION | finite element method | PENALTY | nonlinear Helmholtz equation | POLLUTION | CIP-FEM | error estimates | stability

Journal Article

17.
Full Text
Nonparaxial elliptic waves and solitary waves in coupled nonlinear Helmholtz equations

Communications in Nonlinear Science and Numerical Simulation, ISSN 1007-5704, 10/2016, Volume 39, pp. 134 - 148

•Lamé polynomial ansatz method is developed to construct elliptic wave solutions of coupled nonlinear Helmholtz equations.•Novel superposed elliptic waves and...

Lamé polynomials | Coupled nonlinear Helmholtz system | Jacobi elliptic function | Solitary waves | SCHRODINGER-EQUATIONS | SYSTEM | MATHEMATICS, APPLIED | DARK SOLITONS | PHYSICS, FLUIDS & PLASMAS | BIMODAL OPTICAL-FIBERS | PHYSICS, MATHEMATICAL | Lame polynomials | MULTIMODE | SPECIAL SET | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | KERR MEDIA | PULSE-PROPAGATION | SPATIAL SOLITONS | Beams (radiation) | Mathematical analysis | Nonlinearity | Joining | Mathematical models | Tuning | Constraining

Lamé polynomials | Coupled nonlinear Helmholtz system | Jacobi elliptic function | Solitary waves | SCHRODINGER-EQUATIONS | SYSTEM | MATHEMATICS, APPLIED | DARK SOLITONS | PHYSICS, FLUIDS & PLASMAS | BIMODAL OPTICAL-FIBERS | PHYSICS, MATHEMATICAL | Lame polynomials | MULTIMODE | SPECIAL SET | MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | MECHANICS | KERR MEDIA | PULSE-PROPAGATION | SPATIAL SOLITONS | Beams (radiation) | Mathematical analysis | Nonlinearity | Joining | Mathematical models | Tuning | Constraining

Journal Article

Chaos: An Interdisciplinary Journal of Nonlinear Science, ISSN 1054-1500, 06/2019, Volume 29, Issue 6, p. 063121

We study the formation and propagation of chirped elliptic and solitary waves in the cubic-quintic nonlinear Helmholtz equation. This system describes...

OPTICAL PULSES | MATHEMATICS, APPLIED | TRANSMISSION | STABILITY | MEDIA | DISPERSIVE DIELECTRIC FIBERS | SPATIAL SOLITONS | PHYSICS, MATHEMATICAL | GLASSES | Physics - Pattern Formation and Solitons

OPTICAL PULSES | MATHEMATICS, APPLIED | TRANSMISSION | STABILITY | MEDIA | DISPERSIVE DIELECTRIC FIBERS | SPATIAL SOLITONS | PHYSICS, MATHEMATICAL | GLASSES | Physics - Pattern Formation and Solitons

Journal Article

Journal of Sound and Vibration, ISSN 0022-460X, 10/2013, Volume 332, Issue 21, pp. 5508 - 5522

In this paper, a novel description of periodic solution and homoclinic orbit of undamped Helmholtz—Duffing oscillator is proposed via nonlinear time...

ACOUSTICS | QUADRATIC NONLINEAR OSCILLATORS | PERIODIC-SOLUTIONS | MECHANICS | LINDSTEDT-POINCARE METHOD | EQUATION | ENGINEERING, MECHANICAL

ACOUSTICS | QUADRATIC NONLINEAR OSCILLATORS | PERIODIC-SOLUTIONS | MECHANICS | LINDSTEDT-POINCARE METHOD | EQUATION | ENGINEERING, MECHANICAL

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 04/2019, Volume 60, Issue 4, p. 43508

We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity εl and a complex...

PHYSICS, MATHEMATICAL | Helmholtz equations | Wave propagation | Defocusing | Electromagnetic radiation | Permittivity | Slabs | Continuity (mathematics)

PHYSICS, MATHEMATICAL | Helmholtz equations | Wave propagation | Defocusing | Electromagnetic radiation | Permittivity | Slabs | Continuity (mathematics)

Journal Article

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