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1

New optical solitary waves for unstable Schrödinger equation in nonlinear medium

Zhou Qin, Rezazadeh Hadi, Korkmaz Alper, Eslami Mostafa, Mirzazadeh Mohammad, Rezazadeh Mohammadreza

Optica Applicata

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2019

|

Vol. 49, nr 1

135--150

EN

In this paper and for the first time, we describe and introduce a new extended direct algebraic method which is a new method for solving nonlinear partial differential equations arising in nonlinear optics and nonlinear science. By applying this method, we have constructed new solitary wave solutions for the unstable Schrödinger equation. A large family of traveling wave type exact solutions covering exponential, generalized trigonometric, rational and generalized hyperbolic functions to this equation is determined. The solutions are expressed in explicit forms.

2

Manipulation of the supercontinuum energy distribution based on the positive chirp in photonic crystal fiber

Yang H., Tong X., Zhao S., Huang Y.

Optica Applicata

|

2018

|

Vol. 48, nr 1

75--85

EN

The effect of initial frequency chirp is numerically investigated to obtain high-efficient supercontinuum radiation in photonic crystal fibers with two closely spaced zero-dispersion wavelengths. Our results show that the positive chirp can significantly improve the probability of energy transferred from the soliton to the dispersive wave. And with the increase of the chirp, the energy increases obviously. At the same time, the intensity of the dispersive wave is proportional to the chirp value. Especially, solitons will not appear when the chirp value exceeds 3.9. Therefore, choosing an appropriate positive chirp, we can regulate the energy of the dispersive wave and solitons in photonic crystal fibers.

3

Teoria chaosu jako narzędzie badawcze w naukach o bezpieczeństwie

Snopek M.

Bezpieczeństwo i Technika Pożarnicza

|

2017

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Vol. 47, iss. 3

78--89

PL

Cel: Celem artykułu jest wskazanie obszarów w naukach o bezpieczeństwie, w których może znaleźć zastosowanie teoria chaosu. W pracy ukazano zarówno korzyści, jak i wyzwania związane z omawianym nowym podejściem. Autor podejmuje się również odpowiedzi na pytanie, czy teoria chaosu, zgodnie ze stanowiskiem części naukowców, może być traktowana jako rewolucja naukowa w naukach o bezpieczeństwie. Wprowadzenie: W dobie skomplikowanej sytuacji społeczno-politycznej, stawiającej przed państwami nowe wyzwania, na znaczeniu mogą zyskać poglądy znajdujące się do tej pory poza głównym nurtem nauk o bezpieczeństwie. W celu zapewnienia bezpieczeństwa, zarówno wewnętrznego, jak i międzynarodowego, władze zmuszone są do odejścia od dotychczas stosowanych paradygmatów. Wynika to z tego, że często ich podstawy oparte są na ładzie funkcjonującym w okresie zimnej wojny – postrzegają świat jako podzielony na dwa wrogie sobie obozy. Teoria chaosu pozwala spojrzeć na zagrożenia płynące z terroryzmu, zmian w strukturze etnicznej czy katastrof naturalnych będących następstwem globalnych zmian klimatycznych z nowej perspektywy. Wszystkie wymienione wyżej zagrożenia można próbować wytłumaczyć z zastosowaniem elementów teorii chaosu. Poza funkcją eksplanacyjną, teoria chaosu pełni również funkcję prognostyczną. W sektorze finansowym poszukuje się atraktora odpowiadającego za kurs na giełdzie. Zajmując się tematem zapobiegania niszczycielskim efektom fal tsunami, pod uwagę bierze się już nie tylko metodę historyczną, ale także solitonową konstrukcję tsunami. Należy jednak rozważyć, czy możliwe jest przeniesienie teorii wywodzącej się z gruntu nauk ścisłych do nauk społecznych oraz jakie niesie to ze sobą zagrożenia. Metodologia: W artykule zastosowano metodę analizy, krytyki piśmiennictwa oraz wnioskowanie z doświadczeń i obserwacji. Wnioski: Pomimo nowych możliwości, jakie daje teoria chaosu, uznanie jej za rewolucję w nauce byłoby nadużyciem. Wynika to z fundamentalnej różnicy pomiędzy naukami ścisłymi a społecznymi. W tych drugich badacze w swoich pracach zawsze uwzględniali pewną nieprzewidywalność oraz losowość. Jest to związane z tym, że każda nauka mająca w centrum zainteresowań człowieka, musi brać pod uwagę znaczną liczbę czynników. Sprawia to, że przewidywanie zachowań jednostki, szczególnie w sytuacji kryzysowej, okazuje się niemożliwe. Teoria chaosu dostarcza nam nowych narzędzi do prowadzenia badań. Jej uniwersalność polega na tym, że możemy je stosować zarówno w ramach teorii chaosu, jak i w ramach innych ugruntowanych teorii.

EN

Aim: The aim of this review article is to present possibilities which chaos theory brings into the social sciences. It presents its benefits and challenges that need to be overcome. The author also attempts to answer the question of whether chaos theory can be regarded as a scientific revolution in security studies. Introduction: In the era of complex and turbulent political and social circumstances which pose new challenges to countries in the field of security theories which previously were outside the scientific mainstream can gain in importance. In order to provide domestic and international security, national governments are forced to depart from the paradigms applied so far. One of the important problems, from the perspective of the global situation, is that government strategies were often created with a different world in mind. Many of these derived from the time when the world was divided as a result of the Cold War. Chaos theory allows us to look into the dangers of terrorism, changes in ethnic structure, or global climate change and natural disasters from a new perspective. All these processes can be explained with the use of the elements of chaos theory. In the financial sector, scientists are searching for an attractor which will explain the stock market. Other studies focus on forecasting and preparing for tsunamis based on the soliton theory. However, one of the problems which will be considered in this review article is the possibility of using a theory derived from the exact sciences in the social sciences and the risks that come with it. Methodology: This review article is based on the methods of analysis, critical literature review and deductions stemming from experience and observation. Conclusions: Despite the new opportunities offered by chaos theory, treating it as a revolution in the social sciences would not be warranted. The reason for this is the fundamental difference between the exact and the social sciences. In the latter, researchers always must take into the account certain unpredictability and randomness during studies. This is connected with the fact that every field of science, with human in the centre of its interest, must take into account many diverse factors. Therefore, predicting an individual’s behaviour, especially in a crisis situation, is impossible. Chaos theory, however, provides us with new tools for research in the social sciences. Its universality comes from the fact that it can be used in chaos theory as well as in other theories.

4

Dark and singular optical solitons with competing nonlocal nonlinearities

Zhou Q., Liu L., Zhang H., Mirzazadeh M., Bhrawy A. H., Zerrad E., Moshokoa S., Biswas A.

Optica Applicata

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2016

|

Vol. 46, nr 1

79--86

EN

In this work, we study the dynamics of optical solitons in a synthetic nonlocal nonlinear media. The nonlinear dynamical model which describes the propagation of optical solitons in the weakly nonlocal nonlinear media with parabolic law nonlinearity is investigated analytically. The tool of integration that is the Riccati equation mapping approach is introduced to extract exact traveling wave solutions. As a result, an explicit dark soliton, singular soliton and periodic solutions are derived.

5

Solitary wave solution to the nonlinear evolution equation in cascaded quadratic media beyond the slowly varying envelope approximations

Sarma A K, Biswas A.

Optica Applicata

|

2014

|

Vol. 44, nr 2

205--212

EN

We report an exact bright and dark soliton solution to the nonlinear evolution equation derived by MOSES and WISE (Phys. Rev. Lett. 97, 2006, 073903) for cascaded quadratic media beyond the slowly varying envelope approximations. The integrability aspects of the model are addressed. The travelling wave hypothesis as well as the ansatz method are employed to obtain an exact 1-soliton solution. Both bright and dark soliton solutions are obtained. The corresponding constraint conditions are obtained in order for the soliton solutions to exist.

6

Optical solitons in birefringent fibers with parabolic law nonlinearity

Zhou Q., Zhu Q., Biswas A.

Optica Applicata

|

2014

|

Vol. 44, nr 3

399--409

EN

This paper studies the propagation of optical solitons through birefringent fibers with parabolic law nonlinearity. The Hamiltonian perturbations that are inter-modal dispersion, self-steepening, third-order dispersion and nonlinear dispersions are taken into account. Both, Riccati equation expansion method and Jacobian elliptic equation expansion method are used. Finally, analytical solutions that are Jacobian elliptic periodic traveling wave solutions, periodic solutions, unbounded solutions, singular solutions, bright and dark soliton solutions are obtained under several constraint conditions.

7

The dynamics of interacting solitons for the Korteweg-de Vries equation

Olejniczak M., Błaszak M.

Vibrations in Physical Systems

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2012

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Vol. 25

311--316

EN

The aim of the paper is to give a new insight into the interaction of soliton particles and their dynamics. We introduce the definition of a soliton, soliton particles (interacting solitons) and a theorem about the decomposition of multi-soliton solutions to soliton particles. In the paper we also give a theorem state that the motion of maxima of interacting solitons (in a special case) are roots of fourth order polynomial.

8

Optical solitons with bandwidth limited amplification in non-Kerr law media

Biswas A., Wheeler D.

Optica Applicata

|

2010

|

Vol. 40, nr 4

801--809

EN

This paper studies optical solitons with linear attenuation and bandpass filters. The Kerr and power laws of nonlinearity are considered. The governing equations are integrated by the aid of He's semi-inverse variational principle. The parameter domains are identified.

9

Propagation of Ultrashort Pulses in a Nonlinear Medium

Long Van C., Dinh Thuan B., Goldstein P., Viet Hung N., Hoai Son D.

Computational Methods in Science and Technology

|

2010

|

Vol. spec. iss. (2)

xx-xx

EN

In this paper, using a general propagation equation of ultrashort pulses in an arbitrary dispersive nonlinear medium derived in [8] we study in the specific case of Kerr media. An obtained ultrashort pulse propagation equation which is called Generalized Nonlinear Schrödinger Equation usually has a very complicated form and looking for its solutions is usually a “mission impossible”. Theoretical methods to solve this equation are effective only for some special cases. As an example we describe the method of a developed elliptic Jacobi function expansion. Several numerical methods of finding approximate solutions are simultaneously used. We focus mainly on the following methods: Split-Step, Runge-Kutta and Imaginary-time algorithms. Some numerical experiments are implemented for soliton propagation and interacting high order solitons. We consider also an interesting phenomenon: the collapse of solitons.

10

Modeling Step Index Fiber to Soliton Propagation

Kaczmarek T.

Advances in Electronics and Telecommunications

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2010

|

Vol. 1, no. 2

59--62

EN

Step index fiber modeling process is carried out through numerical solving of eigenvalue equation to calculate propagation constant for fundamental mod. Input data in the process is only index of refraction calculated from Sellmeier dispersive formula for appropriate mol percentage doping of germanium dioxide in silica glass fiber. Output data in the modeling process is optimal value of the normalized frequency, which guarantees that single mode operation region is equal to bright soliton propagation region. Final verification of the process is soliton generation up to sixth-order inside such modeled fiber. In this end nonlinear Schödinger equation is solved numerically for initial condition of hyperbolic secant form. Maximization of single mode operation and bright soliton propagation region is essential in wavelength division multiplexing technique.

11

Linear and nonlinear properties of photonic crystal fibers filled with nematic liquid crystals

Brzdąkiewicz K. A., Laudyn U. A., Karpierz M. A., Woliński T. R., Wójcik J.

Opto-Electronics Review

|

2006

|

Vol. 14, No. 4

287-292

EN

We investigate linear and nonlinear light propagation in the photonic crystal fibers infiltrated with nematic liquid crystals. Such a photonic structure, with periodic modulation of refractive index, which could be additionally controlled by the temperature and by the optical power, allows for the study of discrete optical phenomena. Our theoretical investigations, carried out with the near infrared wavelength of 830 nm, for both focusing and defocusing Kerr-type nonlinearity, show the possibility of the transverse light localization, which can result in the discrete soliton generation. In addition, we present the preliminary experimental results on the linear light propagation in the photonic crystal fiber with the glycerin-water solution and 6CHBT nematics, as the guest materials.

12

Can a variational approach describe pulse splitting in a dispersion managed system?

Infeld E, Matuszewski M, Shino C, Trippenbach M

Optica Applicata

|

2006

|

Vol. 36, nr 4

s. 575--580

EN

When looking for solitons in nonlinear systems, it is often useful to have a simplifying tool. One such tool is the variational method. On the other hand, in the presence of fast oscillations, the wavefunction of the system can split into two distinct parts. This is not describable by the classical variational method. Edwards et al., (J. Phys. B 38(4), 2005, pp. 363–76), introduced a hybrid variational analysis which can describe the dynamics in one selected direction more accurately. However, it remained to be seen how well this method describes the dynamics of solitons, in particular their splitting and subsequent recombining. Here we investigate an application of the hybrid variational analysis to a two dimensional system with dispersion management, where such splitting is known to occur. We conclude that indeed agreement is good. This could encourage wider use of the hybrid method.

13

Nonlinear ship wake waves as a model of rogue waves and a source of danger to the coastal environment: a review

Soomere T.

Oceanologia

|

2006

|

No. 48 (S)

185-202

EN

A substantial part of the energy of wake waves from high-speed ships sailing in shallow water is concentrated in nonlinear components which at times have a solitonic nature. Recent results of investigations into solitonic wave interactions within the framework of the Kadomtsev-Petviashvili equation and their implications for rogue wave theory are reviewed. A surface elevation four times as high as the counterparts occurs if the properties of the interacting waves are specifically balanced. The slope of the water surface may increase eightfold. The resulting structure may persist for a long time. Nonlinear wake components may exert a considerable influence on the marine ecosystem in coastal areas .

14

Wide beam stabilities and instabilities in one dimensional arrays of Kerr-nonlinear channel waveguides

Meier J., Christodoulides D. N., Stegeman G. I., Yang H., Salamo G., Morandotti R., Aitchison J. S., Silberberg Y.

Opto-Electronics Review

|

2005

|

Vol. 13, No. 2

75-84

EN

The propagation of wide (plane-wave like) and narrow high intensity beams at 1550 nm was investigated in 1D arrays of AlGaAs channel waveguides which are nearest-neighbour coupled via evanescent fields. Spatial diffraction (beam spreading) by evanescent field coupling leads to dispersion relations that are periodic with propagation direction and hence exhibit regions of both normal and anomalous diffraction. This results in propagation directions in which filamentation of high intensity beams with superimposed noise occurs, and other regions in which these beams are stable against noise. Both cases were observed experimentally, and the modulation instability gain was measured versus the spatial frequency of the noise. Good agreement with theory was found.

15

Nematic liquid crystal waveguide arrays

Brzdąkiewicz K. A., Karpierz M. A., Fratalocchi A., Assanto G., Nowinowski-Kruszelnicki E.

Opto-Electronics Review

|

2005

|

Vol. 13, No. 2

107-122

EN

We investigate linear and nonlinear light propagation in a voltage-tunable array of waveguide channels in undoped nematic liquid crystals. This novel geometry, based on a photonic structure with a periodic modulation of refractive index controlled by an electric field, offers a wealth of possibilities for the study of discrete optical phenomena. The structure, in conjunction with a giant, non-resonant and voltage-dependent reorientational nonlinearity, allows us to drive the system from bulk diffraction to discrete propagation. Theoretical and experimental investigations, carried out with near infrared light wavelength and powers of a few milliWatts, show the possibility of transverse light localization, resulting in discrete spatial solitons. Such array, with its voltage- and light-adjustable guided-wave confinement and coupling, exhibits potentials for the realization of multifunctional routers and all-optical signal processors with nematic liquid crystals.

16

Arrays of weakly coupled, periodically poled lithium niobate waveguides: beam propagation and discrete spatial quadratic solitons

Iwanow R., Schiek R., Stegeman G., Pertsch T., Lederer F., Min Y., Sohler W.

Opto-Electronics Review

|

2005

|

Vol. 13, No. 2

113-121

EN

Discrete optical systems can be realized as arrays of parallel, weakly coupled, channel waveguides where light normally undergoes "discrete diffraction" via the weak coupling between adjacent channels. Here we describe how light can be forced to maintain a constant field profile on propagation in waveguide arrays, i.e., to localize into a discrete spatial soliton, by using the second order nonlinearity of periodically poled lithium niobate near phase-matching for second harmonic generation. Detailed sample characterization and experimental verification of the excitation of discrete quadratic solitons is reported.

17

Generation of fundamental soliton in the presence of initial linear chirp

Kaczmarek T.

Optica Applicata

|

2004

|

Vol. 34, nr 2

241-248

EN

Generation of fundamental soliton in nonlinear optical fiber from chirped pulses of different initial shapes is discussed. Results of numerical calculations having in view solution of nonlinear Schrödinger equation for complex initial condition using split-step Fourier method are presented. Initial shape-dependent critical value of the chirp parameter is determined. Critical value is such a value of the chirp parameter at which generation of soliton in optical fiber is impossible

18

Exact N-envelope-soliton solutions of the Hirota equation

Shu J.-J.

Optica Applicata

|

2003

|

Vol. 33, nr 2-3

539-546

EN

We discuss some properties of the soliton equations of the type ś u/ś t = S[u, u], where S is a nonlinear operator differential in x, and present the additivity theorems of the class of the soliton equations. On using the theorems, we can construct a new soliton equation through two soliton equations with similar properties. Meanwhile, exact N-envelope-soliton solutions of the Hirota equation are derived through the trace method.

19

Photorefractive materials and solitions

Królikowski W., Kivshar Y., Luther-Davies B.

Opto-Electronics Review

|

2001

|

Vol. 9, No. 3

287-292

EN

We investigated the formation and interaction of bright spatial solitons in photorefractive nonlinear media. We demonstrate solition formation, self-bending, coherent and incoherent collision resulting in energy exchange, solition fusion, birth of new solitions and anomalous repulsion of solitions. The results on formation of multi-comporient incoherent solitions (vector solitions) are also presented.

20

Ultra-short optical pulses having initial chirp

Kaczmarek T.

Journal of Telecommunications and Information Technology

|

2000

|

nr 1-2

59-60

EN

Arbitrary shape optical pulses in nonlinear guides are discussed. The Nonlinear Schrödinger Equation for complex initial conditions is solved numerically using Split-Step Fourier Method and some selected results for solitons are presented. The computations confirm physical expectations of an influence of the chirp magnitude on pulse propagation in nonlinear guide.