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15 pages, 256 KB

Open AccessArticle

Extrinsic Geometry of a Riemannian Manifold and Ricci Solitons

by Ibrahim Al-Dayel and Sharief Deshmukh

Axioms 2025, 14(2), 95; https://doi.org/10.3390/axioms14020095 - 27 Jan 2025

Viewed by 793

The object of this paper is to find a vector field

ξ

and a constant

λ

on an n-dimensional compact Riemannian manifold

(M^{n}, g)

such that we obtain the Ricci soliton

(M^{n}, g, ξ, λ)

. [...] Read more.

The object of this paper is to find a vector field

ξ

and a constant

λ

on an n-dimensional compact Riemannian manifold

(M^{n}, g)

such that we obtain the Ricci soliton

(M^{n}, g, ξ, λ)

. In order to achieve this objective, we choose an isometric embedding provided in the work of Kuiper and Nash in the Euclidean space

(R^{m}, \bar{g})

and choose

ξ

as the tangential component of a constant unit vector on

R^{m}

and call it a Kuiper–Nash vector. If

τ

is the scalar curvature of the compact Riemannian manifold

(M^{n}, g)

with a Kuiper–Nash vector

ξ

, we show that if the integral of the function

ξ (τ)

has a suitable lower bound containing a constant

λ

, then

(M^{n}, g, ξ, λ)

is a Ricci soliton; we call this a Kuiper–Nash Ricci soliton. We find a necessary and sufficient condition involving the scalar curvature

τ

under which a compact Kuiper–Nash Ricci soliton

(M^{n}, g, ξ, λ)

is a trivial soliton. Finally, we find a characterization of an n-dimensional compact trivial Kuiper–Nash Ricci soliton

(M^{n}, g, ξ, λ)

using an upper bound on the integral of

{(d i v ξ)}^{2}

containing the scalar curvature

τ

. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

14 pages, 1877 KB

Open AccessArticle

Robust Soliton Distribution-Based Zero-Watermarking for Semi-Structured Power Data

by Lei Zhao, Yunfeng Zou, Chao Xu, Yulong Ma, Wen Shen, Qiuhong Shan, Shuai Jiang, Yue Yu, Yihan Cai, Yubo Song and Yu Jiang

Electronics 2024, 13(3), 655; https://doi.org/10.3390/electronics13030655 - 4 Feb 2024

Cited by 1 | Viewed by 1828

Abstract

To ensure the security of online-shared power data, this paper adopts a robust soliton distribution-based zero-watermarking approach for tracing semi-structured power data. The method involves extracting partial key-value pairs to generate a feature sequence, processing the watermark into an equivalent number of blocks. [...] Read more.

To ensure the security of online-shared power data, this paper adopts a robust soliton distribution-based zero-watermarking approach for tracing semi-structured power data. The method involves extracting partial key-value pairs to generate a feature sequence, processing the watermark into an equivalent number of blocks. Robust soliton distribution from erasure codes and redundant error correction codes is utilized to generate an intermediate sequence. Subsequently, the error-corrected watermark information is embedded into the feature sequence, creating a zero-watermark for semi-structured power data. In the tracking process, the extraction and analysis of the robust zero-watermark associated with the tracked data facilitate the effective identification and localization of data anomalies. Experimental and simulation validation demonstrates that this method, while ensuring data security, achieves a zero-watermark extraction success rate exceeding 98%. The proposed approach holds significant application value for data monitoring and anomaly tracking in power systems. Full article

(This article belongs to the Special Issue Knowledge Information Extraction Research)

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11 pages, 21773 KB

Open AccessFeature PaperArticle

Swirling of Horizontal Skyrmions into Hopfions in Bulk Cubic Helimagnets

by Andrey O. Leonov

Magnetism 2023, 3(4), 297-307; https://doi.org/10.3390/magnetism3040023 - 19 Oct 2023

Cited by 4 | Viewed by 2968

Abstract

Magnetic hopfions are three-dimensional topological solitons embedded into a homogeneously magnetized background. The internal structure of hopfions is distinguished by the linked preimages—closed loops with a single orientation of the magnetization on the target space

S^{2}

—and is thus characterized by the [...] Read more.

Magnetic hopfions are three-dimensional topological solitons embedded into a homogeneously magnetized background. The internal structure of hopfions is distinguished by the linked preimages—closed loops with a single orientation of the magnetization on the target space

S^{2}

—and is thus characterized by the integer Hopf index

Q_{H}

. Alternatively, hopfions can be visualized as a result of the swirling of two-dimensional bimerons around the direction of an applied magnetic field. Since the bimeron consists of a circular core and an anti-skyrmion crescent, two hopfion varieties can be achieved with either bimeron constituent facing the hopfion interior. In bulk cubic helimagnets, however, the applied magnetic field leads to a spontaneous collapse of hopfions, i.e., the eigen-energy of hopfions has the minimum for zero hopfion radius R. Anti-hopfions with

Q_{H} = - 1

, in this case, pass through the intermediate toron state with two-point defects. Here, we demonstrate that the competing cubic and exchange anisotropies inherent in cubic non-centrosymmetric magnets (e.g., in the Mott insulator Cu

_{2}

OSeO

_{3}

) as a third level of the hierarchy of energy scales following the exchange and Dzyaloshinskii–Moriya interactions, may shift the energy minimum into the region of finite hopfion radii. Full article

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9 pages, 934 KB

Open AccessCommunication

Supersymmetric AdS Solitons, Ground States, and Phase Transitions in Maximal Gauged Supergravity

by Antonio Gallerati

Particles 2023, 6(3), 762-770; https://doi.org/10.3390/particles6030048 - 12 Aug 2023

Viewed by 1521

Abstract

We review some recent soliton solutions in a class of four-dimensional supergravity theories. The latter can be obtained from black hole solutions by means of a double Wick rotation. For special values of the parameters, the new configurations can be embedded in the [...] Read more.

We review some recent soliton solutions in a class of four-dimensional supergravity theories. The latter can be obtained from black hole solutions by means of a double Wick rotation. For special values of the parameters, the new configurations can be embedded in the gauged maximal

N = 8

theory and uplifted in the higher-dimensional

D = 11

theory. We also consider BPS soliton solutions, preserving a certain fraction of supersymmetry. Full article

(This article belongs to the Special Issue Beyond the Standard Models in Particle Physics and Cosmology)

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17 pages, 5085 KB

Open AccessArticle

Embed-Solitons in the Context of Functions of Symmetric Hyperbolic Fibonacci

by Mokhtar. Y. Youssif, Khadeeja A. A. Helal, Manal Yagoub Ahmed Juma, Amna E. Elhag, Abd Elmotaleb A. M. A. Elamin, Mohammed A. Aiyashi and Sayed M. Abo-Dahab

Symmetry 2023, 15(8), 1473; https://doi.org/10.3390/sym15081473 - 25 Jul 2023

Cited by 2 | Viewed by 1661

Abstract

In this article, we discuss the findings of new developments in a class of new triangular functions that blend the quantity functions of the traditional triangular. Considering the significant role played by the triangular functions in applied mathematics, physics, and engineering, it is [...] Read more.

In this article, we discuss the findings of new developments in a class of new triangular functions that blend the quantity functions of the traditional triangular. Considering the significant role played by the triangular functions in applied mathematics, physics, and engineering, it is conceivable to predict that the theory of new triangular functions will provide us with additional interpretations and discoveries in mathematics and physics. The solutions which consider variable separation based on arbitrary functions are constructed to the (3+1)-dimensional Burgers model by presenting the Fibonacci Riccati technique and the linearly independent variable separation approach. This technique’s fundamental concept is to describe the solution of the Burgers model as a polynomial in the Riccati Equation solution that satisfies the symmetrical hyperbolic and triangular Fibonacci functions. Depending on the choice of suitable functions for variable separation, an abundance of new localized solutions were obtained. Moreover, examples such as embedded solitons, rectangle-solitons, plateau-type ring solitons, taper-like solitons, and their interactions with each other, following the symmetrical hyperbolic and triangular Fibonacci functions, as well as the golden mean, could be explored. Full article

(This article belongs to the Section Mathematics)

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11 pages, 1262 KB

Open AccessEditor’s ChoiceArticle

Dynamics and Embedded Solitons of Stochastic Quadratic and Cubic Nonlinear Susceptibilities with Multiplicative White Noise in the Itô Sense

by Zhao Li and Chen Peng

Mathematics 2023, 11(14), 3185; https://doi.org/10.3390/math11143185 - 20 Jul 2023

Cited by 12 | Viewed by 1254

Abstract

The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and [...] Read more.

The main purpose of this paper is to study the dynamics and embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities in the Itô sense, which can further help researchers understand the propagation of soliton nonlinear systems. Firstly, a two-dimensional dynamics system and its perturbation system are obtained by using a traveling wave transformation. Secondly, the phase portraits of the two-dimensional dynamics system are plotted. Furthermore, the chaotic behavior, two-dimensional phase portraits, three-dimensional phase portraits and sensitivity of the perturbation system are analyzed via Maple software. Finally, the embedded solitons of stochastic quadratic and cubic nonlinear susceptibilities are obtained. Moreover, three-dimensional and two-dimensional solitons of stochastic quadratic and cubic nonlinear susceptibilities are plotted. Full article

(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering, 2nd Edition)

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11 pages, 1261 KB

Open AccessArticle

Transmission of Vortex Solitons in Three-Dimensional χ⁽²⁾ Helical-Periodically Poled Ferroelectric Crystals

by Yixi Chen, Aowei Yang, Yangui Zhou, Hexiang He and Jianing Xie

Photonics 2023, 10(7), 818; https://doi.org/10.3390/photonics10070818 - 13 Jul 2023

Viewed by 1716

Abstract

It is well known that bright vortex solitons are unstable in the

χ^{(2)}

nonlinear media due to the strong azimuthal modulation instability. To solve this problem, a quadratic (

χ^{(2)}

)

LiNb O_{3}

ferroelectric crystal with a special kind of [...] Read more.

It is well known that bright vortex solitons are unstable in the

χ^{(2)}

nonlinear media due to the strong azimuthal modulation instability. To solve this problem, a quadratic (

χ^{(2)}

)

LiNb O_{3}

ferroelectric crystal with a special kind of helical-periodically poled structure is proposed. The proposed structure is designed by embedding topological charges into the crystal with a quasi-phase matching technique. Simulation results indicate that vortex solitons containing fundamental-frequency and second-harmonic waves can robustly propagate over a distance. Two types of vortex states are obtained: double vortices state and vortex–antivortex state. The dependence of effective area, propagation constants, and maximum light intensity on the control parameters are presented. These results provide a new solution for robust transmission of bright vortex solitons in a

χ^{(2)}

nonlinear media. Full article

(This article belongs to the Section Optical Interaction Science)

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13 pages, 3815 KB

Open AccessArticle

Solitons Solution of Riemann Wave Equation via Modified Exp Function Method

by Attaullah, Muhammad Shakeel, Bilal Ahmad, Nehad Ali Shah and Jae Dong Chung

Symmetry 2022, 14(12), 2574; https://doi.org/10.3390/sym14122574 - 6 Dec 2022

Cited by 18 | Viewed by 2680

Abstract

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp [...] Read more.

In the areas of tidal and tsunami waves in oceans, rivers, ion and magneto-sound waves in plasmas, electromagnetic waves in transmission lines, homogeneous and stationary media, etc., the Riemann wave equations are attractive nonlinear equations. The modified exp

(- Φ (η))

-function method is used in this article to show how well it can be applied to extract travelling and solitary wave solutions from higher-order nonlinear evolution equations (NLEEs) using the equations mentioned above. Trigonometric, hyperbolic, and exponential functions solitary wave solutions can be extracted using the above-mentioned technique. By changing specific values of the embedded parameters, we can obtain bell-form soliton, consolidated bell-shape soliton, compacton, singular kink soliton, flat kink shape soliton, smooth singular soliton, and other sorts of soliton solutions. The solutions are graphically illustrated in 3D and 2D for the accuracy of the outcome by using the Wolfram Mathematica 10. The verification of numerical solvers on the stability analysis of the solution is substantially aided by the analytic solutions. Full article

(This article belongs to the Special Issue Differential/Difference Equations and Its Application)

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12 pages, 4858 KB

Open AccessArticle

2D BP/InSe Heterostructures as a Nonlinear Optical Material for Ultrafast Photonics

by Yiqing Shu, Zijun Zhong, Chunyang Ma, Penglai Guo, Leiming Wu, Zhitao Lin, Xun Yuan, Jianqing Li, Weicheng Chen and Quanlan Xiao

Nanomaterials 2022, 12(11), 1809; https://doi.org/10.3390/nano12111809 - 25 May 2022

Cited by 16 | Viewed by 2957

Abstract

The BP/InSe heterojunction has attracted the attention of many fields in successful combined high hole mobility of black phosphorus (BP) and high electron mobility of indium selenide (InSe), and enhanced the environmental stability of BP. Nevertheless, photonics research on the BP/InSe heterostructure was [...] Read more.

The BP/InSe heterojunction has attracted the attention of many fields in successful combined high hole mobility of black phosphorus (BP) and high electron mobility of indium selenide (InSe), and enhanced the environmental stability of BP. Nevertheless, photonics research on the BP/InSe heterostructure was insufficient, while both components are considered promising in the field. In this work, a two-dimensional (2D) BP/InSe heterostructure was fabricated using the liquid-phase exfoliation method. Its linear and non-linear optical (NLO) absorption was characterized by ultraviolet−visible−infrared and Open-aperture Z-scan technology. On account of the revealed superior NLO properties, an SA based on 2D BP/InSe was prepared and embedded into an erbium-doped fiber laser, traditional soliton pulses were observed at 1.5 μm with the pulse duration of 881 fs. Furthermore, harmonic mode locking of bound solitons and dark-bright soliton pairs were also obtained in the same laser cavity due to the cross-coupling effect. The stable mode-locked operation can be maintained for several days, which overcome the low air stability of BP. This contribution further proves the excellent optical properties of 2D BP/InSe heterostructure and provides new probability of developing nano-photonics devices for the applications of double pulses laser source and long-distance information transmission. Full article

(This article belongs to the Special Issue Xene-Related Nanostructures for Versatile Applications)

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10 pages, 2639 KB

Open AccessArticle

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

by Sheng Zhang and Bo Xu

Mathematics 2022, 10(3), 486; https://doi.org/10.3390/math10030486 - 2 Feb 2022

Cited by 4 | Viewed by 2330

Abstract

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund [...] Read more.

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term. Full article

(This article belongs to the Special Issue Partial Differential Equations with Applications: Analytical Methods)

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11 pages, 1199 KB

Open AccessArticle

Light Confinement with Structured Beams in Gold Nanoparticle Suspensions

by Argelia Balbuena Ortega, Felix E. Torres-González, Valentin López Gayou, Raul Delgado Macuil, Gaetano Assanto and Karen Volke-Sepulveda

Photonics 2021, 8(6), 221; https://doi.org/10.3390/photonics8060221 - 15 Jun 2021

Cited by 3 | Viewed by 3707

Abstract

We carry out an experimental campaign to investigate the nonlinear self-defocusing propagation of singular light beams with various complex structures of phase and intensity in a colloidal suspension of gold nanoparticles with a plasmonic resonance near the laser wavelength (532nm). Studying optical vortices [...] Read more.

We carry out an experimental campaign to investigate the nonlinear self-defocusing propagation of singular light beams with various complex structures of phase and intensity in a colloidal suspension of gold nanoparticles with a plasmonic resonance near the laser wavelength (532nm). Studying optical vortices embedded in Gaussian beams, Bessel vortices and Bessel-cosine (necklace) beams, we gather evidence that while intense vortices turn into two-dimensional dark solitons, all structured wavepackets are able to guide a weak Gaussian probe of different wavelength (632.8 nm) along the dark core. The probe confinement also depends on the topological charge of the singular pump. Full article

(This article belongs to the Special Issue Singular Optics)

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18 pages, 4336 KB

Open AccessArticle

Highly Directive Biconic Antennas Embedded in a Dielectric

by Alessandro Chiolerio, Lorenzo Diazzi and Daniele Funaro

Appl. Sci. 2020, 10(24), 8828; https://doi.org/10.3390/app10248828 - 10 Dec 2020

Cited by 3 | Viewed by 2770

Abstract

Designing antennas suitable for generating highly directive electromagnetic signals has become a fundamental task. This is particularly relevant for the development of efficient and sustainable point-to-point communication channels, and for energy transfer. Indeed, these are nowadays expanding areas of research. In order to [...] Read more.

Designing antennas suitable for generating highly directive electromagnetic signals has become a fundamental task. This is particularly relevant for the development of efficient and sustainable point-to-point communication channels, and for energy transfer. Indeed, these are nowadays expanding areas of research. In order to deal with said particular wave phenomena, an extension of the electrodynamics equations is taken into account, where exact solitonic type solutions are admitted. These waves may have compact support and travel along a straight line, without dissipation, at the speed of light. The result suggests the design of biconic type antennas having specific properties that are numerically examined in this paper. The cones, supplied with an oscillating source, are embedded in a dielectric material of suitable shape, with the purpose of driving the signal in the proper direction. The computations based on the extended model are aimed toward simulating the possibility of generating peculiar wave behaviors, in view of practical implementations in the framework of point-to-point communications or wireless power transmission. Full article

(This article belongs to the Special Issue 10th Anniversary of Applied Sciences: Invited Papers in Electrical, Electronics and Communications Engineering Section)

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10 pages, 1047 KB

Open AccessArticle

Construction of Solitary Two-Wave Solutions for a New Two-Mode Version of the Zakharov-Kuznetsov Equation

by Imad Jaradat and Marwan Alquran

Mathematics 2020, 8(7), 1127; https://doi.org/10.3390/math8071127 - 10 Jul 2020

Cited by 59 | Viewed by 2867

Abstract

A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are [...] Read more.

A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are used to obtain exact bell-shaped soliton solutions and singular soliton solutions to the proposed model. Two-dimensional and three-dimensional plots are also provided to illustrate the interaction dynamics of the obtained two-wave exact solutions upon increasing the phase-velocity parameter. Full article

(This article belongs to the Section E: Applied Mathematics)

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24 pages, 749 KB

Open AccessReview

Singular Mean-Field States: A Brief Review of Recent Results

by Elad Shamriz, Zhaopin Chen, Boris A. Malomed and Hidetsugu Sakaguchi

Condens. Matter 2020, 5(1), 20; https://doi.org/10.3390/condmat5010020 - 20 Mar 2020

Cited by 15 | Viewed by 3470

Abstract

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model [...] Read more.

This article provides a focused review of recent findings which demonstrate, in some cases quite counter-intuitively, the existence of bound states with a singularity of the density pattern at the center; the states are physically meaningful because their total norm converges. One model of this type is based on the 2D Gross–Pitaevskii equation (GPE), which combines the attractive potential

\sim r^{- 2}

and the quartic self-repulsive nonlinearity, induced by the Lee–Huang–Yang effect (quantum fluctuations around the mean-field state). The GPE demonstrates suppression of the 2D quantum collapse, driven by the attractive potential, and emergence of a stable ground state (GS), whose density features an integrable singularity

\sim r^{- 4 / 3}

at

r \to 0

. Modes with embedded angular momentum exist too, but they are unstable. A counter-intuitive peculiarity of the model is that the GS exists even if the sign of the potential is reversed from attraction to repulsion, provided that its strength is small enough. This peculiarity finds a relevant explanation. The other model outlined in the review includes 1D, 2D, and 3D GPEs, with the septimal (seventh-order), quintic, and cubic self-repulsive terms, respectively. These equations give rise to stable singular solitons, which represent the GS for each dimension D, with the density singularity

\sim r^{- 2 / (4 - D)}

. Such states may be considered the results of screening a “bare” delta-functional attractive potential by the respective nonlinearities. Full article

(This article belongs to the Special Issue Fluctuations and Highly Non-linear Phenomena in Superfluids and Superconductors II)

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33 pages, 5912 KB

Open AccessArticle

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

by Jorge Fujioka, Alfredo Gómez-Rodríguez and Áurea Espinosa-Cerón

Appl. Sci. 2017, 7(4), 340; https://doi.org/10.3390/app7040340 - 30 Mar 2017

Cited by 1 | Viewed by 3890

Abstract

We study linear and nonlinear pulse propagation models whose linear dispersion relations present bands of forbidden frequencies or forbidden wavenumbers. These bands are due to the interplay between higher-order dispersion and one of the terms (a second-order derivative with respect to the propagation [...] Read more.

We study linear and nonlinear pulse propagation models whose linear dispersion relations present bands of forbidden frequencies or forbidden wavenumbers. These bands are due to the interplay between higher-order dispersion and one of the terms (a second-order derivative with respect to the propagation direction) which appears when we abandon the slowly varying envelope approximation. We show that as a consequence of these forbidden bands, narrow pulses radiate in a novel and peculiar way. We also show that the nonlinear equations studied in this paper have exact soliton-like solutions of different forms, some of them being embedded solitons. The solutions obtained (of the linear as well as the nonlinear equations) are interesting since several arguments suggest that the Cauchy problems for these equations are ill-posed, and therefore the specification of the initial conditions is a delicate issue. It is also shown that some of these equations are related to elliptic curves, thus suggesting that these equations might be related to other fields where these curves appear, such as the theory of modular forms and Weierstrass ℘ functions, or the design of cryptographic protocols. Full article

(This article belongs to the Special Issue Guided-Wave Optics)

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