WEAKLY NONPARAXIAL EFFECTS ON THE PROPAGATION OF (1+1)D SPATIAL SOLITONS IN INHOMOGENEOUS KERR MEDIA

The widely-used approach to study the beam propagation in Kerr media is based on the slowly varying envelope approximation (SVEA) which is also known as the paraxial approximation. Within this approximation, the beam evolution is described by the nonlinear Schrödinger (NLS) equation. In this paper, we extend the NLS equation by including higher-order terms to study the effects of nonparaxiality on the soliton propagation in inhomogeneous Kerr media. The result is still a one-way wave equation which means that all back-reflections are neglected. The accuracy of this approximation exceeds the standard SVEA. By performing several numerical simulations, we show that the NLS equation produces reasonably good predictions for relatively small degrees of nonparaxiality, as expected. However, in the regions where the envelope beam is changing rapidly as in the breakup of a multisoliton bound state, the nonparaxiality plays an important role.

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On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽

10.3390/sym13050878 ◽

2021 ◽

Vol 13 (5) ◽

pp. 878

Author(s):

Alexei Cheviakov ◽

Denys Dutykh ◽

Aidar Assylbekuly

Keyword(s):

Wave Equation ◽

Conservation Laws ◽

Numerical Simulations ◽

Solitary Waves ◽

Local Symmetry ◽

Symmetry And Conservation Laws ◽

Higher Order ◽

Special Equation ◽

Long Wave ◽

Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

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Higher-order anisotropic models in phase separation

Advances in Nonlinear Analysis ◽

10.1515/anona-2016-0137 ◽

2017 ◽

Vol 8 (1) ◽

pp. 278-302 ◽

Cited By ~ 5

Author(s):

Laurence Cherfils ◽

Alain Miranville ◽

Shuiran Peng

Keyword(s):

Phase Separation ◽

Numerical Simulations ◽

Global Attractor ◽

Higher Order ◽

Well Posedness ◽

Anisotropic Models ◽

Higher Order Terms

Abstract Our aim in this paper is to study higher-order (in space) Allen–Cahn and Cahn–Hilliard models. In particular, we obtain well-posedness results, as well as the existence of the global attractor. We also give, for the Allen–Cahn models, numerical simulations which illustrate the effects of the higher-order terms and the anisotropy.

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Beam Propagation Factor of Hollow Higher-order Cosh-Gaussian Beams

10.21203/rs.3.rs-1036932/v1 ◽

2021 ◽

Author(s):

Faroq Saad ◽

Ahmed Abdulrab Ali Ebrahim ◽

Abdelmajid Belafhal

Keyword(s):

Mathematical Model ◽

Numerical Simulations ◽

Beam Propagation ◽

General Characteristic ◽

Higher Order ◽

Beam Waist ◽

Gaussian Beams ◽

Propagation Factor ◽

Analytical Formulae ◽

Beam Propagation Factor

Abstract Based on the second-order moments definition, we investigate in this paper the beam propagation factor of new mathematical model of Hollow higher-order Cosh-Gaussian (HhCG) beams. Two analytical formulae of the M2-factor of HhCG beams are derived. Moreover, numerical simulations are developed to illustrate the effects of the beams orders n and l, the parameter δ and the beam waist ω0 on the M2-factor. The result shows a more general characteristic of higher-order Cosh-Gaussian, Cosh-Gaussian and the fundamental Gaussian beams can be obtained as specials cases of HhCG beams.

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Diffraction by a spherical cap

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100002140 ◽

1963 ◽

Vol 59 (1) ◽

pp. 197-209 ◽

Cited By ~ 6

Author(s):

D. P. Thomas

Keyword(s):

Wave Equation ◽

Steady State ◽

Systematic Approach ◽

Low Frequency ◽

Higher Order ◽

Wave Number ◽

Systematic Method ◽

Spherical Cap ◽

Typical Dimension ◽

Higher Order Terms

1. It was first observed by Rayleigh(17) in 1897 that low-frequency approximations to the solutions of the steady-state wave equation could be obtained from the solutions of the corresponding static problems. Rayleigh determined only the first term in the expansion of the solution in powers of the product of the wave number and a typical dimension. Many recent investigations have been concerned with deriving a systematic method of calculating the higher-order terms in the above expansion. Most of the problems which have been solved in this manner have been concerned with scattering by a disk or by a strip: the first systematic approach to these problems was that of Bouwkamp ((3), p. 71).

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Asymptotic analysis of the crack tip stress field (consideration of higher order terms)

Сибирский журнал вычислительной математики ◽

10.15372/sjnm20190307 ◽

2019 ◽

Keyword(s):

Stress Field ◽

Asymptotic Analysis ◽

Crack Tip ◽

Higher Order ◽

Higher Order Terms ◽

Crack Tip Stress

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Higher Order Approximation for Stochastic Space Fractional Wave Equation Forced by an Additive Space-Time Gaussian Noise

Journal of Scientific Computing ◽

10.1007/s10915-021-01415-0 ◽

2021 ◽

Vol 87 (1) ◽

Author(s):

Xing Liu ◽

Weihua Deng

Keyword(s):

Wave Equation ◽

Gaussian Noise ◽

Order Approximation ◽

Higher Order ◽

Space Time ◽

Fractional Wave Equation ◽

Stochastic Space

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A unified approach to the Timoshenko 3D beam-column element tangent stiffness matrix considering higher-order terms in the strain tensor and large rotations

International Journal of Solids and Structures ◽

10.1016/j.ijsolstr.2021.02.014 ◽

2021 ◽

Vol 222-223 ◽

pp. 111003

Author(s):

Marcos Antonio Campos Rodrigues ◽

Rodrigo Bird Burgos ◽

Luiz Fernando Martha

Keyword(s):

Stiffness Matrix ◽

Strain Tensor ◽

Higher Order ◽

Unified Approach ◽

Tangent Stiffness ◽

Tangent Stiffness Matrix ◽

Large Rotations ◽

Higher Order Terms ◽

Column Element

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Using the Multi-Parameter Fracture Mechanics for more Accurate Description of Stress/Displacement Crack Tip Fields

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.586.237 ◽

2013 ◽

Vol 586 ◽

pp. 237-240 ◽

Cited By ~ 8

Author(s):

Lucie Šestáková

Keyword(s):

Stress Distribution ◽

Fracture Mechanics ◽

Boundary Conditions ◽

Displacement Field ◽

Singular Term ◽

Crack Tip ◽

Higher Order ◽

Precise Description ◽

Accurate Knowledge ◽

Higher Order Terms

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.

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