Showing papers on "Quintic function published in 2022"
TL;DR: In this article , a sub-equation method is implemented to construct exact solutions for the conformable perturbed nonlinear Schrödinger equation, and the order of the expected polynomial-type solution is obtained using the homogeneous balancing approach.
Abstract: The sub-equation method is implemented to construct exact solutions for the conformable perturbed nonlinear Schrödinger equation. In this paper, we consider three different types of nonlinear perturbations: The quadratic–cubic law, the quadratic–quartic–quintic law, and the cubic–quintic–septic law. The properties of the conformable derivative are discussed and applied with the help of a suitable wave transform that converts the governing model to a nonlinear ordinary differential equation. Furthermore, the order of the expected polynomial-type solution is obtained using the homogeneous balancing approach. Dark and singular soliton solutions are derived.
38 citations
TL;DR: In this paper , some novel soliton wave solutions are obtained and formulated in distinct forms such as rational, hyperbolic, and trigonometric forms to show the physical and dynamical behavior of the nonparaxial pulse propagation.
Abstract: This article investigates some novel solitary wave solutions of the cubic-quintic nonlinear Helmholtz equation by implementing two recent analytical (Khater II and generalized exponential methods) techniques. As a result of seeing the nonparaxial effect, this model was developed to show how a pulse propagates in a planar waveguide with Kerr-like and quintic nonlinearities as well as spatial dispersion. When the traditional approximation of a slowly changing envelope fails, this effect takes over. Some novel soliton wave solutions are obtained and formulated in distinct forms such as rational, hyperbolic, and trigonometric forms. The obtained solutions are represented through various figures’ types to show the physical and dynamical behavior of the nonparaxial pulse propagation. The model’s features are demonstrated, such as its intriguing bright, anti-bright, periodic, singular, gray, chirped anti-dark, and dark solitary waves depending upon the nature of nonlinearities. The solutions’ originality is discussed by comparing it with those constructed in recently published articles. All built results are checked their accuracy by putting them back into the original model by employing Mathematica 12.
29 citations
TL;DR: In this article , the cubic-quintic nonlinear Schrödinger equation (CQ-NLSE) was used to describe the propagation properties of nonlinear periodic waves (PW) in an optical fiber.
Abstract: In this paper, we study the cubic-quintic nonlinear Schrödinger equation (CQ-NLSE) to describe the propagation properties of nonlinear periodic waves (PW) in an optical fiber . We find chirped periodic waves (CPW) with some Jacobi elliptic functions (JEF). We also obtain some solitary waves (SW) like dark, bright, hyperbolic and singular solitons. The chirp that corresponds to each of these optical solitons is also determined. The pair intensity is shown to be related to the nonlinear chirp, which is determined by self-frequency shift and pause self-steepening (SS). The shape of profile for these waves will also be display.
21 citations
TL;DR: The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation as discussed by the authors , and two simplified formulations are suggested.
Abstract: The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. This paper gives a systematical analysis of the formulation, and two simplified formulations are suggested. The cubic-quintic Duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. It can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory.
19 citations
TL;DR: In this article , exact solutions from perturbed complex Ginzburg-Landau equation with Kerr law and cubic-quintic-septic nonlinearity were obtained by using the trial equation method and complete discriminant system for polynomial method.
Abstract: This paper secures exact solutions from perturbed complex Ginzburg–Landau equation that is taken into account with Kerr law and cubic–quintic–septic nonlinearity. Two approaches are used, namely the trial equation method and complete discriminant system for polynomial method. The abundant exact solutions obtained can better analyze the complex optical phenomena and further demonstrate their essence.
18 citations
17 citations
Book•
10 Jan 2022
TL;DR: In this article, a unified proof for n = 3, 4, and 5 was given based on the parametrizations of low rank rings due to Bhargava and Delone-Faddeev.
Abstract: A fascinating tale of mayhem, mystery, and mathematics. Attached to each degree n number field is a rank n−1 lattice called its shape. This thesis shows that the shapes of Sn-number fields (of degree n = 3, 4, or 5) become equidistributed as the absolute discriminant of the number field goes to infinity. The result for n = 3 is due to David Terr. Here, we provide a unified proof for n = 3, 4, and 5 based on the parametrizations of low rank rings due to Bhargava and Delone–Faddeev. We do not assume any of those words make any kind of sense, though we do make certain assumptions about how much time the reader has on her hands and what kind of sense of humor she has.
17 citations
TL;DR: A review of results for solitons in 2D and 3D models of nonlinear dissipative media can be found in this paper , where stable fundamental and vortex DSs are presented in the framework of the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity, which combines linear and quintic loss with cubic gain.
Abstract: This article offers a review of results for solitons in 2D and 3D models of nonlinear dissipative media. The existence of such solitons requires to maintain two balances: between nonlinear self-focusing and linear diffraction and/or dispersion, and between loss and gain. Due to these conditions, dissipative solitons (DSs) exist not in families, but as isolated solutions. The main issue is stability of 2D and 3D DSs, especially vortical ones. First, stable 2D DSs are presented in the framework of the complex Ginzburg-Landau equation with the cubic-quintic (CQ) nonlinearity, which combines linear and quintic loss with cubic gain. In addition to fundamental (zero-vorticity) DSs, stable spiral DSs are presented too, with vorticities 1 and 2. Stable 2D solitons were also found in a system of two linearly-coupled fields, with linear gain acting in one and linear loss in the other. In this case, the cubic loss (without quintic terms) is sufficient for the stability of fundamental and vortex DSs. In addition to truly localized states, weakly localized ones are presented too, in a model with nonlinear loss without explicit gain, the losses being compensated by influx of power from infinity. Other classes of 2D models which are considered here use spatially modulated loss or gain to predict many species of robust DSs, including ones featuring complex periodically recurring metamorphoses. Stable fundamental and vortex solitons are also produced by models with a trapping or spatially periodic potential. In the latter case, 2D gap DSs are considered. Further, 2D dissipative models with spin-orbit coupling give rise to stable semi-vortices, with vorticity carried by one component. Along with the 2D solitons, the review includes 3D fundamental and vortex DSs, stabilized by the CQ nonlinearity and/or external potentials, and collisions between them.
15 citations
TL;DR: In this article , the Hirota bilinear method was used to study soliton interactions of quasi-1D Bose-Einstein Condensate system (BECs) with dipole-dipole attraction and repulsion.
Abstract: With the help of the Hirota bilinear method (HBM), we study soliton interactions of quasi-1D Bose-Einstein Condensate system (BECs) with dipole-dipole attraction and repulsion. BEC is an extended form of nonlinear Schrödinger equation (NLSE) and it consists of quadratic-cubic nonlinearities, linear gain or loss and time modulated dispersion. Due to its spatially varying coefficients property it has significance in the field of fluid dynamics, classical and quantum field theories, nonlinear optics and physics etc. We will also discuss soliton interactions with graphically descriptions for QDNLSE. We obtain some parabolic, anti parabolic, M-shaped, W-shaped, butterflies, bright, anti dark, V-shaped, S-shaped and other solitons for our governing models.
14 citations
TL;DR: In this paper , the higher-order nonlinear Schrödinger equation (NLSE) with third order dispersion, fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms is presented.
Abstract: In this paper, we present the higher-order nonlinear Schrödinger equation (NLSE) with third order dispersion (3OD), fourth-order dispersion (4OD), and cubic-quintic nonlinearity (CQNL) terms that define the propagation of ultrashort pulses. Two analytical methods, which are the new Kudryashov’s method and the unified Riccati equation expansion method, are implemented to extract the analytical soliton solutions of the presented equation for the first time. Thus, bright, dark, and singular soliton solutions are acquired. To illustrate the physical behavior of some of the obtained solutions, 3D, 2D, and contour graphs are depicted. In particular, to understand the effects of the group velocity dispersion, 3OD, 4OD, CQNLs, self-steepening coefficient terms, and group velocity term of the traveling wave transformation on the soliton dynamics of the proposed equation, 2D plots for different values of coefficients are represented. The obtained results provide us with the knowledge that the presented model can be examined from a physical perspective. It can be concluded that the used methods are effective approaches to derive the analytical solutions for the NLSE.
12 citations
TL;DR: In this paper , the authors recover dark and bright cubic-quartic optical solitons with the cubic-quintic-septic-nonic nonlinear Schrödinger equation.
TL;DR: In this paper , the Khater II analytical technique is used to examine novel soliton structures for the fractional nonlinear third-order Schrödinger (3-FNLS) problem.
Abstract: In this paper, the Khater II analytical technique is used to examine novel soliton structures for the fractional nonlinear third-order Schrödinger (3-FNLS) problem. The 3-FNLS equation explains the dynamical behavior of a system’s quantum aspects and ultra-short optical fiber pulses. Additionally, it determines the wave function of a quantum mechanical system in which atomic particles behave similarly to waves. For example, electrons, like light waves, exhibit diffraction patterns when passing through a double slit. As a result, it was fair to suppose that a wave equation could adequately describe atomic particle behavior. The correctness of the solutions is determined by comparing the analytical answers obtained with the numerical solutions and determining the absolute error. The trigonometric Quintic B-spline numerical (TQBS) technique is used based on the computed required criteria. Analytical and numerical solutions are represented in a variety of graphs. The strength and efficacy of the approaches used are evaluated.
TL;DR: In this paper , a generalized derivative nonlinear Schrödinger equation with varying dispersion, self-steepening effect, cubic-quintic nonlinearity, and gain or loss was modeled in femtosecond light propagation in an optical fiber.
Abstract: We demonstrate new types of chirped self-similar waves for a generalized derivative nonlinear Schrödinger equation with varying dispersion, self–steepening effect, cubic–quintic nonlinearity, and gain or loss. The equation arises in modeling femtosecond light propagation in an optical fiber with spatial parameter variations. The newly found self-similar structures take the shape of gray and kink pulses. It is observed that the frequency chirp accompanying these self-similar pulses depends crucially on the intensity of the wave and its amplitude can be effectively governed by adjusting the self-steepening parameter. The dynamical behaviors of the chirped self-similar gray and kink waves are analyzed in a periodic distributed amplification system. The acquired self-similar structures display rich dynamical evolutions that are important in practical applications.
TL;DR: In this paper , a multi-objective trajectory planning approach based on an improved elitist non-dominated sorting genetic algorithm (INSGA-II) is proposed, where a new composite polynomial that by combining of quintic polynomials with cubic Bezier curves is proposed.
Abstract: Abstract Robot manipulators perform a point-point task under kinematic and dynamic constraints. Due to multi-degree-of-freedom coupling characteristics, it is difficult to find a better desired trajectory. In this paper, a multi-objective trajectory planning approach based on an improved elitist non-dominated sorting genetic algorithm (INSGA-II) is proposed. Trajectory function is planned with a new composite polynomial that by combining of quintic polynomials with cubic Bezier curves. Then, an INSGA-II, by introducing three genetic operators: ranking group selection (RGS), direction-based crossover (DBX) and adaptive precision-controllable mutation (APCM), is developed to optimize travelling time and torque fluctuation. Inverted generational distance, hypervolume and optimizer overhead are selected to evaluate the convergence, diversity and computational effort of algorithms. The optimal solution is determined via fuzzy comprehensive evaluation to obtain the optimal trajectory. Taking a serial-parallel hybrid manipulator as instance, the velocity and acceleration profiles obtained using this composite polynomial are compared with those obtained using a quintic B-spline method. The effectiveness and practicability of the proposed method are verified by simulation results. This research proposes a trajectory optimization method which can offer a better solution with efficiency and stability for a point-to-point task of robot manipulators.
TL;DR: In this paper , the moduli space M of semistable objects of Mukai vector 2λ1+2λ2 in the Kuznetsov component of Y admits a symplectic resolution M˜, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady.
TL;DR: In this paper , three well known properties of solitons, like Painleve test (P -test), Hirota bilinear method (HBM) and extended modified auxiliary equation mapping (EMAEM) in SCQNM, were discussed.
TL;DR: In this article , a class of physically interesting PT-symmetric δ(x)-csch potentials containing three types of potentials, namely, csch-coth potential, δ-csch, and csch coth potentials are studied.
17 Aug 2022
TL;DR: In this article , the authors investigated the Eckhaus instability of stationary patterns arising in hyperbolic reaction diffusion models on large finite domains, in both supercritical and subcritical regime, and deduced the cubic and cubic-quintic real Ginzburg-Landau equations ruling the evolution of pattern amplitude close to criticality.
Abstract: Abstract We have theoretically investigated the phenomenon of Eckhaus instability of stationary patterns arising in hyperbolic reaction–diffusion models on large finite domains, in both supercritical and subcritical regime. Adopting multiple-scale weakly-nonlinear analysis, we have deduced the cubic and cubic–quintic real Ginzburg–Landau equations ruling the evolution of pattern amplitude close to criticality. Starting from these envelope equations, we have provided the explicit expressions of the most relevant dynamical features characterizing primary and secondary quantized branches of any order: stationary amplitude, existence and stability thresholds and linear growth rate. Particular emphasis is given on the subcritical regime, where cubic and cubic–quintic Ginzburg–Landau equations predict qualitatively different dynamical pictures. As an illustrative example, we have compared the above-mentioned analytical predictions to numerical simulations carried out on the hyperbolic modified Klausmeier model, a conceptual tool used to describe the generation of stationary vegetation stripes over flat arid environments. Our analysis has also allowed to elucidate the role played by inertia during the transient regime, where an unstable patterned state evolves towards a more favorable stable configuration through sequences of phase-slips. In particular, we have inspected the functional dependence of time and location at which wavelength adjustment takes place as well as the possibility to control these quantities, independently of each other.
TL;DR: In this article , the authors considered the large data scattering problem for the 2D and 3D cubic-quintic nonlinear Schrödinger equation in the focusing-focusing regime.
TL;DR: In this paper , the optical solitons of nonlinear Schrödinger equation with cubic-quintic law nonlinearity, in the presence of self-frequency shift and self-steepening, have been studied.
Abstract: In this manuscript, the optical solitons of nonlinear Schrödinger equation (NLSE) with cubic–quintic law nonlinearity, in the presence of self-frequency shift and self-steepening, has been studied. The ultrahigh capacity propagation and transit of laser light pulses in optical fibres were described using this form of equation. To extract new results, two strong methodologies has been used. To extract the exact solution unified method has been employed. The solutions obtained by this analytical method, are in form of polynomial and rational function solution. Moreover, the validity of non-singular solutions has guaranteed by a limitation condition that is graphically illustrated in 3D. The 2D graphical representation are also used to demonstrate the influence of parameters on the predicted non-singular solutions. The other technique, used for qualitative analysis, is bifurcation. The system has been transformed into a planer dynamical system, which has been transformed into a hamiltonian system. All the possible phase portrait has been plotted by complete discrimination method. The acquired results are novel and have not been recorded before and they indicate that the proposed methodologies may be used to investigate innovative soliton solutions and phase portraits for any NLSE. • In this paper, aim is to find new forms of soliton solutions related to NLSE model with cubic-quintic nonlinearity. • Unified strategy has been used to find new solutions in the forms of rational and polynomial functions. • Qualitative analysis, using planer dynamical system is also carried out of the model. • All the obtained results are depicted graphically for some specific values of parameters.
TL;DR: In this article , a generalized nonlinear Schrödinger equation with pseudo-quintic nonlinearity and self-steepening effect is proposed to propagate ultrashort pulses at least a few tens of optical cycles in duration through a negative index material.
TL;DR: In this article , an approximate solution to the damped quintic-cubic nonlinear Duffing-Mathieu equation via a modified homotopy perturbation method (HPM) is presented.
Abstract: Abstract This paper is devoted to obtain an approximate solution to the damped quintic–cubic nonlinear Duffing–Mathieu equation via a modified homotopy perturbation method (HPM). The modification under consideration deals with the improvement of the HPM with the exponential decay parameter. This scheme allows us to get a solution to the damped nonlinear Duffing–Mathieu equation, which the classical HPM failed to obtain. It is found that the solutions and the characteristic curves are affected by the presence of the damping force. The frequency-amplitude characteristics of a symbiotic solution are confirmed as well as the stability condition is carried out in the (non)-resonance cases. All the calculations are done via Mathematica. The comparison between both of the numerical and analytical solutions showed a very good agreement. Illustrated graphs are plotted for a superior realization of periodic motions in the Duffing–Mathieu oscillator. Nonlinear behaviors of each oscillation motion have been characterized through frequency curves.
05 Oct 2022
TL;DR: In this paper , the authors used the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators to study other conservative and non-conservative damped nonlinear oscillators.
Abstract: Oscillators are omnipresent; most of them are inherently nonlinear. Though a nonlinear equation mostly does not yield an exact analytic solution for itself, plethora of elementary yet practical techniques exist for extracting important information about the solution of equation. The purpose of this chapter is to introduce some new techniques for the readers which are carefully illustrated using mainly the examples of Duffing’s oscillator. Using the exact analytical solution to cubic Duffing and cubic-quinbic Duffing oscillators, we describe the way other conservative and some non conservative damped nonlinear oscillators may be studied using analytical techniques described here. We do not make use of perturbation techniques. However, some comparison with such methods are performed. We consider oscillators having the form x¨+fx=0 as well as x¨+2εẋ+fx=Ft, where x=xt and f=fx and Ft are continuous functions. In the present chapter, sometimes we will use f−x=−fx and take the approximation fx≈∑j=1Npjxj, where j=1,3,5,⋯N only odd integer values and x∈−AA. Moreover, we will take the approximation fx≈∑j=0Npjxj, where j=1,2,3,⋯N, and x∈−AA. Arbitrary initial conditions are considered. The main idea is to approximate the function f=fx by means of some suitable cubic or quintic polynomial. The analytical solutions are expressed in terms of the Jacobian and Weierstrass elliptic functions. Applications to plasma physics, electronic circuits, soliton theory, and engineering are provided.
TL;DR: In this article , diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach.
Abstract: This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.