Search results for: sets of coupled nonlinear differential equation

Authors: Mohammad Reza Shahnazari, Reza Kazemi, Ali Saberi

Abstract:

Most of the engineering problems are in nonlinear forms. Nonlinear boundary layer problems defined in infinite intervals contain specific complexities, especially in boundary layer condition conformance. As an example of these nonlinear complex problems, the well-known Blasius equation can be mentioned, which itself is one of the classic boundary layer problems. No analytical solution has been proposed yet for the Blasius equation due to its complexity. In this paper, an analytical method, namely the Kourosh method, based on the singularity perturbation method and the Liao homotopy analysis is utilized to solve the Blasius problem. In this method, an inner solution is developed in the [0,1] interval to expedite the solution convergence. The magnitude of the f ˝(0), as an essential quantity for determining the physical parameters, is directly calculated from the solution of the boundary condition problem. The advantages of this solution are that it does not need any numerical solution, it has a closed form and that its validation is shown in the entire [0,∞] interval. Furthermore, all of the desirable parameters could be extracted through a series of simple analytical operations from the final solution. This solution also satisfies the continuity conditions, which is one of the main contributions of this paper in comparison with most of the other proposed analytical solutions available in the literature. Comparison with numerical solutions reveals that the proposed method is highly accurate and convenient for application.

Keywords: Blasius equation, boundary layer, Kourosh method, analytical solution

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Authors: Prashant Pandey

Abstract:

In the present article, an effective Laguerre collocation method is used to obtain the approximate solution of a system of coupled fractional-order non-linear reaction-advection-diffusion equation with prescribed initial and boundary conditions. In the proposed scheme, Laguerre polynomials are used together with an operational matrix and collocation method to obtain approximate solutions of the coupled system, so that our proposed model is converted into a system of algebraic equations which can be solved employing the Newton method. The solution profiles of the coupled system are presented graphically for different particular cases. The salient feature of the present article is finding the stability analysis of the proposed method and also the demonstration of the lower variation of solute concentrations with respect to the column length in the fractional-order system compared to the integer-order system. To show the higher efficiency, reliability, and accuracy of the proposed scheme, a comparison between the numerical results of Burger’s coupled system and its existing analytical result is reported. There are high compatibility and consistency between the approximate solution and its exact solution to a higher order of accuracy. The exhibition of error analysis for each case through tables and graphs confirms the super-linearly convergence rate of the proposed method.

Keywords: fractional coupled PDE, stability and convergence analysis, diffusion equation, Laguerre polynomials, spectral method

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Authors: Arcady Ponosov

Abstract:

Many well-known age-structured population models are derived from the celebrated McKendrick-von Foerster equation (MFE), also called the biological conservation law. A similar technique is suggested for the stochastically perturbed MFE. This technique is shown to produce stochastic versions of the deterministic population models, which appear to be very different from those one can construct by simply appending additive stochasticity to deterministic equations. In particular, it is shown that stochastic Nicholson’s blowflies model should contain both additive and multiplicative stochastic noises. The suggested transformation technique is similar to that used in the deterministic case. The difference is hidden in the formulas for the exact solutions of the simplified boundary value problem for the stochastically perturbed MFE. The analysis is also based on the theory of stochastic delay differential equations.

Keywords: boundary value problems, population models, stochastic delay differential equations, stochastic partial differential equation

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Authors: Alexander Blokhin, Ekaterina Kruglova, Boris Semisalov

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Mathematical model based on the mesoscopic theory of polymer dynamics is developed for numerical simulation of the flows of polymeric liquid between two coaxial cylinders. This model is a system of nonlinear partial differential equations written in the cylindrical coordinate system and coupled with the heat conduction equation including a specific dissipation term. The stationary flows similar to classical Poiseuille ones are considered, and the resolving equations for the velocity of flow and for the temperature are obtained. For solving them, a fast pseudospectral method is designed based on Chebyshev approximations, that enables one to simulate the flows through the channels with extremely small relative values of the radius of inner cylinder. The numerical analysis of the dependance of flow on this radius and on the values of dissipation constant is done.

Keywords: dynamics of polymeric liquid, heat dissipation, singularly perturbed problem, pseudospectral method, Chebyshev polynomials, stabilization technique

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Authors: Jawad Zakir, Syed Irtiza Ali Shah, Rana Shaharyar, Sidra Mahmood

Abstract:

Y-force equation comprises aerodynamic forces, drag and side force with side slip angle β and weight component along with the coupled roll (φ) and pitch angles (θ). This research deals with the linearization of Y-force equation using Small Disturbance theory assuming equilibrium flight conditions for different state variables of aircraft. By using assumptions of Small Disturbance theory in non-linear Y-force equation, finally reached at linearized lateral rigid body equation of motion; which says that in linearized Y-force equation, the lateral acceleration is dependent on the other different aerodynamic and propulsive forces like vertical tail, change in roll rate (Δp) from equilibrium, change in yaw rate (Δr) from equilibrium, change in lateral velocity due to side force, drag and side force components due to side slip, and the lateral equation from coupled rotating frame to decoupled rotating frame. This paper describes implementation of this lateral linearized equation for aircraft control systems. Another significant parameter considered on which y-force equation depends is ‘c’ which shows that any change bought in the weight of aircrafts wing will cause Δφ and cause lateral force i.e. Y_c. This simplification also leads to lateral static and dynamic stability. The linearization of equations is required because much of mathematics control system design for aircraft is based on linear equations. This technique is simple and eases the linearization of the rigid body equations of motion without using any high-speed computers.

Keywords: Y-force linearization, small disturbance theory, side slip, aerodynamic force drag, lateral rigid body equation of motion

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Authors: C. Pislaru, A. Shebani

Abstract:

This paper uses the radial basis function neural network (RBFNN) for system identification of nonlinear systems. Five nonlinear systems are used to examine the activity of RBFNN in system modeling of nonlinear systems; the five nonlinear systems are dual tank system, single tank system, DC motor system, and two academic models. The feed forward method is considered in this work for modelling the non-linear dynamic models, where the K-Means clustering algorithm used in this paper to select the centers of radial basis function network, because it is reliable, offers fast convergence and can handle large data sets. The least mean square method is used to adjust the weights to the output layer, and Euclidean distance method used to measure the width of the Gaussian function.

Keywords: system identification, nonlinear systems, neural networks, radial basis function, K-means clustering algorithm

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Authors: H. Kazemi Esfeh, V. Akbari, M. Ehdaei, T. N. G. Borhani, A. Shamiri, M. Najafi

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In an industrial fluidized bed polymerization reactor, particle size distribution (PSD) plays a significant role in the reactor efficiency evaluation. The computational fluid dynamic (CFD) models coupled with population balance equation (CFD-PBM) have been extensively employed to investigate the flow behavior in the poly-disperse multiphase fluidized bed reactors (FBRs) utilizing ANSYS Fluent code. In this study, an existing CFD-PBM/ DQMOM coupled modeling framework has been used to highlight its potential to analyze the industrial-scale gas phase polymerization reactor. The predicted results reveal an acceptable agreement with the observed industrial data in terms of pressure drop and bed height. The simulated results also indicate that the higher particle growth rate can be achieved for bigger particles. Hence, the 2D CFD-PBM/DQMOM coupled model can be used as a reliable tool for analyzing and improving the design and operation of the gas phase polymerization FBRs.

Keywords: computational fluid dynamics, population balance equation, fluidized bed polymerization reactor, direct quadrature method of moments

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Authors: Fakhreddin Abedi, Wah June Leong

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Exponential stability of stochastic differential equations with non trivial solutions is provided in terms of Lyapunov functions. The main result of this paper establishes that, under certain hypotheses for the dynamics f(.) and g(.), practical exponential stability in probability at the small neighborhood of the origin is equivalent to the existence of an appropriate Lyapunov function. Indeed, we establish exponential stability of stochastic differential equation when almost all the state trajectories are bounded and approach a sufficiently small neighborhood of the origin. We derive sufficient conditions for exponential stability of stochastic differential equations. Finally, we give a numerical example illustrating our results.

Keywords: exponential stability in probability, stochastic differential equations, Lyapunov technique, Ito's formula

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Authors: S. S. P. M. Isa, N. M. Arifin, R. Nazar, N. Bachok, F. M. Ali, I. Pop

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A theoretical study has been presented to describe the boundary layer flow and heat transfer on an exponentially shrinking sheet with a variable wall temperature and suction, in the presence of magnetic field. The governing nonlinear partial differential equations are converted into ordinary differential equations by similarity transformation, which are then solved numerically using the shooting method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented through graphs and tables for several sets of values of the parameters. The effects of the governing parameters on the flow and heat transfer characteristics are thoroughly examined.

Keywords: exponentially shrinking sheet, magnetic field, mixed convection, suction

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Authors: Minas Balyan

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Nowadays X-ray nonlinear diffraction and nonlinear effects are investigated due to the presence of the third generation synchrotron sources and XFELs. X-ray third order nonlinear dynamical diffraction is considered as well. Using the nonlinear model of the usual visible light optics the third-order nonlinear Takagi’s equations for monochromatic waves and the third-order nonlinear time-dependent dynamical diffraction equations for X-ray pulses are obtained by the author in previous papers. The obtained equations show, that even if the Fourier-coefficients of the linear and the third order nonlinear susceptibilities are zero (forbidden reflection), the dynamical diffraction in the nonlinear case is related to the presence in the nonlinear equations the terms proportional to the zero order and the second order nonzero Fourier coefficients of the third order nonlinear susceptibility. Thus, in the third order nonlinear Bragg diffraction case a nonlinear analogue of the well-known Renninger effect takes place. In this work, the 'third order nonlinear Renninger effect' is considered theoretically.

Keywords: Bragg diffraction, nonlinear Takagi’s equations, nonlinear Renninger effect, third order nonlinearity

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Authors: O. Acan, Y. Keskin

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In this study, we will carry out a comparative study between the reduced differential transform method, the adomian decomposition method, the variational iteration method and the homotopy analysis method. These methods are used in many fields of engineering. This is been achieved by handling a kind of 2-Dimensional and forth-order partial differential equations called the Kuramoto–Sivashinsky equations. Three numerical examples have also been carried out to validate and demonstrate efficiency of the four methods. Furthermost, it is shown that the reduced differential transform method has advantage over other methods. This method is very effective and simple and could be applied for nonlinear problems which used in engineering.

Keywords: reduced differential transform method, adomian decomposition method, variational iteration method, homotopy analysis method

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Authors: O. Boussoufi, K. Lamrini Uahabi, M. Atounti

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The main purpose of this paper is to calculate the fractal dimension of some Julia Sets and Mandelbrot Set in the Misiurewicz Points. Using Matlab to generate the Julia Sets images that match the Misiurewicz points and using a Fractal software, we were able to find different measures that characterize those fractals in textures and other features. We are actually focusing on fractal dimension and the error calculated by the software. When executing the given equation of regression or the log-log slope of image a Box Counting method is applied to the entire image, and chosen settings are available in a FracLAc Program. Finally, a comparison is done for each image corresponding to the area (boundary) where Misiurewicz Point is located.

Keywords: box counting, FracLac, fractal dimension, Julia Sets, Mandelbrot Set, Misiurewicz Points

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Authors: Ogunrinde Roseline Bosede

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This paper presents the development, analysis and implementation of an inverse polynomial numerical method which is well suitable for solving initial value problems in first order ordinary differential equations with applications to sample problems. We also present some basic concepts and fundamental theories which are vital to the analysis of the scheme. We analyzed the consistency, convergence, and stability properties of the scheme. Numerical experiments were carried out and the results compared with the theoretical or exact solution and the algorithm was later coded using MATLAB programming language.

Keywords: differential equations, numerical, polynomial, initial value problem, differential equation

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Authors: G. Akgun, I. Algul, H. Kurtaran

Abstract:

In this paper geometrically nonlinear static behavior of laminated composite hollow super-elliptic beams is investigated using generalized differential quadrature method. Super-elliptic beam can have both oval and elliptic cross-sections by adjusting parameters in super-ellipse formulation (also known as Lamé curves). Equilibrium equations of super-elliptic beam are obtained using the virtual work principle. Geometric nonlinearity is taken into account using von-Kármán nonlinear strain-displacement relations. Spatial derivatives in strains are expressed with the generalized differential quadrature method. Transverse shear effect is considered through the first-order shear deformation theory. Static equilibrium equations are solved using Newton-Raphson method. Several composite super-elliptic beam problems are solved with the proposed method. Effects of layer orientations of composite material, boundary conditions, ovality and ellipticity on bending behavior are investigated.

Keywords: generalized differential quadrature, geometric nonlinearity, laminated composite, super-elliptic cross-section

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Authors: Ayaz Ahmad

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TOPIC: A study of the formation ,existness and stability of localised pulses in pde Ayaz Ahmad ,NITP, Abstract:In this paper we try to govern the evolution deterministic variable over space and time .We analysis the behaviour of the model which allows us to predict and understand the possible behaviour of the physical system .Bifurcation theory provides a basis to systematically investigate the models for invariant sets .Exploring the behaviour of PDE using bifurcation theory which provides many challenges both numerically and analytically. We use the derivation of a non linear partial differential equation which may be written in this form ∂u/∂t+c ∂u/∂x+∈(∂^3 u)/(∂x^3 )+¥u ∂u/∂x=0 We show that the temperature increased convection cells forms. Through our work we look for localised solution which are characterised by sudden burst of aeroidic spatio-temporal evolution. Key word: Gaussian pulses, Aeriodic ,spatio-temporal evolution ,convection cells, nonlinearoptics, Dr Ayaz ahmad Assistant Professor Department of Mathematics National institute of technology Patna ,Bihar,,India 800005 [email protected] +91994907553

Keywords: Gaussian pulses, aeriodic, spatio-temporal evolution, convection cells, nonlinear optics

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Authors: Abdulatif Abdusalam, Mohamed Shaban

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In this paper, we consider the nonlinear pulse propagation through a nonuniform birefringent fiber Bragg grating (FBG) whose index modulation depth varies along the propagation direction. Here, the pulse propagation is governed by the nonlinear birefringent coupled mode (NLBCM) equations. To form the Bragg soliton outside the photonic bandgap (PBG), the NLBCM equations are reduced to the well known NLS type equation by multiple scale analysis. As we consider the pulse propagation in a nonuniform FBG, the pulse propagation outside the PBG is governed by inhomogeneous NLS (INLS) rather than NLS. We, then, discuss the formation of soliton in the FBG known as Bragg soliton whose central frequency lies outside but close to the PBG of the grating structure. Further, we discuss Bragg soliton compression due to a delicate balance between the SPM and the varying grating induced dispersion. In addition, Bragg soliton collision, Bragg soliton switching and possible logic gates have also been discussed.

Keywords: Bragg grating, non uniform fiber, non linear pulse

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Authors: Melusi Khumalo, Anastacia Dlamini

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In this paper, we consider a numerical solution for nonlinear Fredholm integral equations of the second kind. We work with uniform mesh and use the Lagrange polynomials together with the Galerkin finite element method, where the weight function is chosen in such a way that it takes the form of the approximate solution but with arbitrary coefficients. We implement the finite element method to the nonlinear Fredholm integral equations of the second kind. We consider the error analysis of the method. Furthermore, we look at a specific example to illustrate the implementation of the finite element method.

Keywords: finite element method, Galerkin approach, Fredholm integral equations, nonlinear integral equations

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Authors: Hui-Yun Cao, Hai-Qing Zhou

Abstract:

The high precision measurements and experiments play more and more important roles in particle physics and atomic physics. To analyse the precise experimental data sets, the corresponding precise and reliable theoretical calculations are necessary. Until now, the form factors of elemental constituents such as pion and proton are still attractive issues in current Quantum Chromodynamics (QCD). In this work, the two-photon-exchange (TPE) effects in ep→enπ⁺ at small -t are discussed within a hadronic model. Under the pion dominance approximation and the limit mₑ→0, the TPE contribution to the amplitude can be described by a scalar function. We calculate TPE contributions to the amplitude, and the unpolarized differential cross section with the only elastic intermediate state is considered. The results show that the TPE corrections to the unpolarized differential cross section are about from -4% to -20% at Q²=1-1.6 GeV². After considering the TPE corrections to the experimental data sets of unpolarized differential cross section, we analyze the TPE corrections to the separated cross sections σ(L,T,LT,TT). We find that the TPE corrections (at Q²=1-1.6 GeV²) to σL are about from -10% to -30%, to σT are about 20%, and to σ(LT,TT) are much larger. By these analyses, we conclude that the TPE contributions in ep→enπ⁺ at small -t are important to extract the separated cross sections σ(L,T,LT,TT) and the electromagnetic form factor of π⁺ in the experimental analysis.

Keywords: differential cross section, form factor, hadronic, two-photon

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Authors: K. N. S. Kasi Viswanadham

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Coupled system of second order linear and nonlinear boundary value problems occur in various fields of Science and Engineering. In the formulation of the problem, any one of 81 possible types of boundary conditions may occur. These 81 possible boundary conditions are written as a combination of four boundary conditions. To solve a coupled system of boundary value problem with these converted boundary conditions, a collocation method with cubic B-splines as basis functions has been developed. In the collocation method, the mesh points of the space variable domain have been selected as the collocation points. The basis functions have been redefined into a new set of basis functions which in number match with the number of mesh points in the space variable domain. The solution of a non-linear boundary value problem has been obtained as the limit of a sequence of solutions of linear boundary value problems generated by quasilinearization technique. Several linear and nonlinear boundary value problems are presented to test the efficiency of the proposed method and found that numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

Keywords: collocation method, coupled system, cubic b-splines, mesh points

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Authors: Seyed Amin Vakili, Sahar Sadat Vakili, Seyed Ehsan Vakili, Nader Abdoli Yazdi

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Differential equations are of fundamental importance in engineering and applied mathematics, since many physical laws and relations appear mathematically in the form of such equations. The equilibrium state of structures consisting of one-dimensional elements can be described by an ordinary differential equation. The response of these kinds of structures under the loading, namely relationship between the displacement field and loading field, can be predicted by the solution of these differential equations and on satisfying the given boundary conditions. When the effect of change of geometry under loading is taken into account in modeling of equilibrium state, then these differential equations are partially integrable in quartered. They also exhibit instability characteristics when the structures are loaded compressively. The purpose of this paper is to represent the ability of the Modified Newmark Method in analyzing flexural-torsional instability of struts for both bifurcation and non-bifurcation structural systems. The results are shown to be very accurate with only a small number of iterations. The method is easily programmed, and has the advantages of simplicity and speeds of convergence and easily is extended to treat material and geometric nonlinearity including no prismatic members and linear and nonlinear spring restraints that would be encountered in frames. In this paper, these abilities of the method will be extended to the system of linear differential equations that govern strut flexural torsional stability.

Keywords: instability, torsion, flexural, buckling, modified newmark method stability

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Authors: G. S. Thakur

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In this paper, four new operators (O1, O2, O3, O4) are proposed, defined and considered to study the new properties and identities on hesitant fuzzy sets. These operators are useful for different operation on hesitant fuzzy sets. The various theorems are proved using the new operators. The study of the proposed new operators has opened a new area of research and applications.

Keywords: vague sets, hesitant fuzzy sets, intuitionistic fuzzy set, fuzzy sets, fuzzy multisets

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Authors: Muath Awadalla, Maen Awadallah

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Banking sector covers different activities including lending money to customers. However, it is commonly known that customers pay money they have borrowed including an added amount called interest. Compound interest rate is an approach used in determining the interest to be paid. The instant compounded amount to be paid by a debtor is obtained through a differential equation whose main parameters are the rate and the time. The rate used by banks in a country is often defined by the government of the said country. In Switzerland, for instance, a negative rate was once applied. In this work, a new approach of modeling the compound interest is proposed using Hadamard fractional derivative. As a result, it appears that depending on the fraction value used in derivative the amount to be paid by a debtor might either be higher or lesser than the amount determined using the classical approach.

Keywords: compound interest, fractional differential equation, hadamard fractional derivative, optimization

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Authors: K. P. Mredula, D. C. Vakaskar

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The article traces developments and evolution of various algorithms developed for solving partial differential equations using the significant combination of wavelet with few already explored solution procedures. The approach depicts a study over a decade of traces and remarks on the modifications in implementing multi-resolution of wavelet, finite difference approach, finite element method and finite volume in dealing with a variety of partial differential equations in the areas like plasma physics, astrophysics, shallow water models, modified Burger equations used in optical fibers, biology, fluid dynamics, chemical kinetics etc.

Keywords: multi-resolution, Haar Wavelet, partial differential equation, numerical methods

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Authors: A. Abdikian

Abstract:

Propagation of nonlinear acoustic wave in dense electron-positron (e-p) plasmas in the presence of an external magnetic field and stationary ions (to neutralize the plasma background) is studied. By means of the quantum hydrodynamics model and applying the reductive perturbation method, the Zakharov-Kuznetsov equation is derived. Using the bifurcation theory of planar dynamical systems, the compressive structure of electrostatic solitary wave and periodic travelling waves is found. The numerical results show how the ion density ratio, the ion cyclotron frequency, and the direction cosines of the wave vector affect the nonlinear electrostatic travelling waves. The obtained results may be useful to better understand the obliquely nonlinear electrostatic travelling wave of small amplitude localized structures in dense magnetized quantum e-p plasmas and may be applicable to study the particle and energy transport mechanism in compact stars such as the interior of massive white dwarfs etc.

Keywords: bifurcation theory, phase portrait, magnetized electron-positron plasma, the Zakharov-Kuznetsov equation

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Authors: Sachin Kumar

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Fuzzy fractional diffusion equation is widely useful to depict different physical processes arising in physics, biology, and hydrology. The motive of this article is to deal with the fuzzy fractional diffusion equation. We study a mathematical model of fuzzy space-time fractional diffusion equation in which unknown function, coefficients, and initial-boundary conditions are fuzzy numbers. First, we find out a fuzzy operational matrix of Legendre polynomial of Caputo type fuzzy fractional derivative having a non-singular Mittag-Leffler kernel. The main advantages of this method are that it reduces the fuzzy fractional partial differential equation (FFPDE) to a system of fuzzy algebraic equations from which we can find the solution of the problem. The feasibility of our approach is shown by some numerical examples. Hence, our method is suitable to deal with FFPDE and has good accuracy.

Keywords: fractional PDE, fuzzy valued function, diffusion equation, Legendre polynomial, spectral method

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Authors: K. Meera Saheb, K. Krishna Bhaskar

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Complex structures used in many fields of engineering are made up of simple structural elements like beams, plates etc. These structural elements, sometimes carry concentrated masses at discrete points, and when subjected to severe dynamic environment tend to vibrate with large amplitudes. The frequency amplitude relationship is very much essential in determining the response of these structural elements subjected to the dynamic loads. For Timoshenko beams, the effects of shear deformation and rotary inertia are to be considered to evaluate the fundamental linear and nonlinear frequencies. A commonly used method for solving vibration problem is energy method, or a finite element analogue of the same. In the present Coupled Displacement Field method the number of undetermined coefficients is reduced to half when compared to the famous Rayleigh Ritz method, which significantly simplifies the procedure to solve the vibration problem. This is accomplished by using a coupling equation derived from the static equilibrium of the shear flexible structural element. The prime objective of the present paper here is to study, in detail, the effect of a central concentrated mass on the large amplitude free vibrations of uniform shear flexible beams. Accurate closed form expressions for linear frequency parameter for uniform shear flexible beams with a central concentrated mass was developed and the results are presented in digital form.

Keywords: coupled displacement field, coupling equation, large amplitude vibrations, moderately thick plates

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Authors: Mokhtar Labiod, Nabil Ikhlef, Keltoum Bouherine, Olivier Leroy

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A numerical model for the radio-frequency (RF) Argon discharge chamber is developed to simulate the low pressure low temperature inductively coupled plasma. This model will be of fundamental importance in the design of the plasma magnetic control system. Electric and magnetic fields inside the discharge chamber are evaluated by solving a magnetic vector potential equation. To start with, the equations of the ideal magnetohydrodynamics theory will be presented describing the basic behaviour of magnetically confined plasma and equations are discretized with finite element method in cylindrical coordinates. The discharge chamber is assumed to be axially symmetric and the plasma is treated as a compressible gas. Plasma generation due to ionization is added to the continuity equation. Magnetic vector potential equation is solved for the electromagnetic fields. A strong dependence of the plasma properties on the discharge conditions and the gas temperature is obtained.

Keywords: direct-coupled model, magnetohydrodynamic, modelling, plasma torch simulation

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Authors: L. A. Ostrovsky, Yu. A. Stepanyants

Abstract:

Effect of Earth rotation on nonlinear waves is a practically important and theoretically challenging problem of fluid mechanics and geophysics. Whereas the large-scale, geostrophic processes such as Rossby waves are a classical object of oceanic and atmospheric physics, rotation effects on mesoscale waves are not well studied. In particular, the Coriolis force can radically modify the behavior of nonlinear internal gravity waves in the ocean having spatial scales of 1-10 kilometers and time durations of few hours. In the last decade, such a non-trivial behavior was observed more than once. Similar effects are possible for magnetic sound in the ionosphere. Here we outline the main physical peculiarities in the behavior of nonlinear internal waves due to the rotation effect and present some results of our recent studies. The consideration is based on the fourth-order equation derived by one of the authors as a rotation-modified Korteweg–de Vries (rKdV) equation which includes two types of dispersion: one is responsible for the finiteness of depth as in the classical KdV equation; another is due to the Coriolis effect. This equation is, in general, non-integrable; moreover, under the conditions typical of oceanic waves (positive dispersion parameter), it does not allow solitary solutions at all. In the opposite case (negative dispersion) which is possible for, e.g., magnetic sound, solitary solutions do exist and can form complex bound states (multisoliton). Another non-trivial properties of nonlinear internal waves with rotation include, to name a few, the ‘terminal’ damping of the initial KdV soliton disappearing in a finite time due to radiation losses caused by Earth’s rotation, and eventual transformation of a KdV soliton into a wave packet (an envelope soliton). The new results to be discussed refer to the interaction of a soliton with a long background wave. It is shown, in particular, that in this case internal solitons can exist since the radiation losses are compensated by energy pumping from the background wave. Finally, the relevant oceanic observations of rotation effect on internal waves are briefly described.

Keywords: Earth rotation, internal waves, nonlinear waves, solitons

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Authors: Suchi Barua, Narottam Das, Sven Nordholm, Mohammad Razaghi

Abstract:

This paper presents nonlinear pulse propagation characteristics for different input optical pulse shapes with various input pulse energy levels in semiconductor optical amplifiers. For simulation of nonlinear pulse propagation, finite-difference beam propagation method is used to solve the nonlinear Schrödinger equation. In this equation, gain spectrum dynamics, gain saturation are taken into account which depends on carrier depletion, carrier heating, spectral-hole burning, group velocity dispersion, self-phase modulation and two photon absorption. From this analysis, we obtained the output waveforms and spectra for different input pulse shapes as well as for different input energies. It shows clearly that the peak position of the output waveforms are shifted toward the leading edge which due to the gain saturation of the SOA for higher input pulse energies. We also analyzed and compared the normalized difference of full-width at half maximum for different input pulse shapes in the SOA.

Keywords: finite-difference beam propagation method, pulse shape, pulse propagation, semiconductor optical amplifier

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Authors: Jade Martínez-Llinàs, Pere Colet

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By exploring the possible dynamical regimes in a prototypical model for mutually delay-coupled OEOs, here it is shown that two mutually coupled non-identical OEOs, besides in- and out-of-phase square-waves, can generate stable square-wave pulses synchronized at a quarter of the period (T/4) in a broad parameter region. The key point to obtain T/4 solutions is that the two OEO operate with mixed feedback, namely with negative feedback in one and positive in the other. Furthermore, the coexistence of multiple solutions provides a large degree of flexibility for tuning the frequency in the GHz range without changing any parameter. As a result the two coupled OEOs system is good candidate to be implemented for information encoding as a high-capacity memory device.

Keywords: nonlinear optics, optoelectronic oscillators, square waves, synchronization

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