Quintic Nonlinear Terms Research Articles

Modulational instability and chirped modulated wave, chirped optical solitons for a generalized (3+1)-dimensional cubic-quintic medium with self-frequency shift and self-steepening nonlinear terms

We consider a generalized (3+1)-dimensional nonlinear Schrödinger with cubic-quintic nonlinear and self-frequency shift and self-steepening terms. We disrupt the plane wave to study the stability of the wave in this media. We examine how various factors—such as the amplitude of the plane wave, cubic and quintic nonlinear terms, self-steepening, and self-frequency shift—affect the modulational instability (MI). Our findings reveal that the amplitude of the plane wave and the quintic nonlinear term can expand the MI bands and increase the amplitude of the MI growth rate. Conversely, the self-steepening and self-frequency shift terms exert opposite effects, narrowing the MI bands and reducing the amplitude of the MI growth rate. We investigate the existence of modulated chirped rational and polynomial Jacobi elliptic function solutions and chirped optical solitons.

High fidelity control of a many-body Tonks-Girardeau gas with an effective mean-field approach

Shortcuts to adiabaticity (STA) are powerful tools that can be used to control quantum systems with high fidelity. They work particularly well for single particle and noninteracting systems which can be described exactly and which possess invariant or self-similar dynamics. However, finding an exact STA for strongly correlated many-body systems can be difficult, as their complex dynamics may not be easily described, especially for larger systems that do not possess self-similar solutions. Here, we design STAs for one-dimensional bosonic gas in the Tonks-Girardeau limit by using a mean-field approach that succinctly captures the strong interaction effects through a quintic nonlinear term in the Schrödinger equation. We show that for the case of the harmonic oscillator with a time-dependent trap frequency the mean-field approach works exactly and recovers the well-known STA from literature. To highlight the robustness of our approach we also show that it works effectively for anharmonic potentials, achieving higher fidelities than other typical control techniques. Published by the American Physical Society 2024

Impact of weak nonlocal nonlinearity on modulational instability and chirp-free soliton trains in parabolic law optical fiber

This study considers the generation and evolution of chirp-free soliton trains in a focusing cubic-quintic nonlinear optical fiber media (by utilizing experimental parameters), under the influence of weak nonlocal nonlinearity. Analysis of modulational instability (MI) reveals that the MI gain exist only in the anomalous dispersion regime, with the quintic nonlinear term increasing the maximum gain and sideband frequencies. The nonlocal nonlinearity parameter generally suppresses the impact of MI, because the maximum gain is greatly reduced with increase in nonlocality. By using the subsidiary ordinary differential equation method, stationary periodic soliton trains are obtained; with the intensity of the optical beam greatly reduced and frequency increased as nonlocality is gradually stepped up. However, an increase in nonlocality instead reduces the frequency of the traveling optical signals; derived via an efficient transformation method. Our results strongly suggest that nonlocal nonlinearity has potential applications in the control of laser beams; as nonlinear periodic optical signals can be easily transformed to very weak quasi-plane waves.

Exact periodic wave solutions of the cubic-quintic Zakharov equation and their evolution with Hamilton energy

In this paper, we study the exact periodic wave solutions of the Zakharov equation with cubic and quintic nonlinear terms, and their evolution with the energy of Hamiltonian system corresponding to the amplitudes. Based on the theory of plane dynamical system, we first make a detailed qualitative analysis to the plane dynamical system corresponding to the amplitudes of traveling wave solutions of the studied equation, then by applying the analysis method based on the first integral and several appropriate transformations, all seven families of elliptic function periodic wave solutions of the Zakharov equation are obtained. In addition, by studying the evolution limit of periodic wave solutions with respect to Hamilton energy and using the analysis method based on the first integral, all ten pairs of solitary wave solutions of the studied equations are also given under various parameter conditions. From the evolution analysis to the periodic wave solutions with respect to Hamilton energy, it can be seen that it is the energy H of the Hamiltonian system corresponding to the studied equation taking values in different ranges that makes the traveling wave solution of this equation appear as periodic wave solution or solitary wave solution.

The fourth-order dispersion effect on the soliton waves and soliton stabilities for the cubic-quintic Gross–Pitaevskii equation

A generalized nonlinear Gross–Pitaevskii equation with the third-, fourth-order dispersions, and the cubic, quintic nonlinear terms is considered, which shows that the wave velocity, amplitude, inverse temporal width and wave number of propagating waves are uniquely dependent on the parameters of second-, third- and fourth-order dispersions. We discuss the anomalous dispersion relations of the cubic-quintic Gross–Pitaevskii(CQGP) equation with second-, third- and fourth-order dispersions, and find that the fourth-order dispersion has a significant influence on the characteristics of the obtained soliton wave solutions. In particular, we analyze the effect of the fourth-order dispersion function r(t) on periodic, bright and dark wave solutions of the CQGP model. We report several families of non-autonomous soliton solutions with different amplitude surfaces, and see that the fourth-order dispersion is essential in a broad range of real physical media as it influences the wave dynamics. Some numerical simulations of bright and dark solitons for the CQGP equation with the different external potentials are considered, which shows the external potential plays a certain stabilizing role for the coexistence state of bright and dark solitons. These dynamic analyses play a significant role in improving the dispersive transport of the nonlinear wave in parabolic-law media.

An Investigation of Dynamical Behavior of a Wing Model

AbstractBifurcations of equilibria of a wing model have been investigated in this paper. It is shown that the quintic nonlinear terms in the pitch and the plunge coordinate have affected the bifurcation structure of nontrivial equilibria in different degree. In contrast with the quintic stiffening parameter in plunge, the quintic parameter in pitch has a relatively significant effect, which will affect the number, position and stability of nontrivial equilibria. Therein two pairs of nontrivial equilibria with opposite stability coexist or disappear by two fold bifurcations. If the freestream velocity has been taken as a continuation parameter, it will affect the bifurcation structure of all the equilibria, including the trivial and the nontrivial. Wherein the equilibria vary from a trivial to two nontrivial ones by a pitchfork bifurcation. Then one of nontrivial equilibria experiences a supercritical Hopf bifurcation and the bifurcated limit cycles form an ellipsoidal structure with the limit cycles bifurcated from the trivial equilibrium.

Double and triple-pole solutions for the third-order flow equation of the Kaup-Newell system with zero/nonzero boundary conditions

In this work, the double and triple-pole solutions for the third-order flow equation of Kaup-Newell system (TOFKN) with zero boundary conditions (ZBCs) and non-zero boundary conditions (NZBCs) are investigated by means of the Riemann-Hilbert (RH) approach stemming from the inverse scattering transformation. Starting from spectral problem of the TOFKN, the analyticity, symmetries, asymptotic behavior of the Jost function and scattering matrix, the matrix RH problem with ZBCs and NZBCs are constructed. Then the obtained RH problem with ZBCs and NZBCs can be solved in the case of scattering coefficients with double or triple zeros, and the reconstruction formula of potential, trace formula as well as theta condition are also derived correspondingly. Specifically, the general formulas of N-double and N-triple poles solutions with ZBCs and NZBCs are derived systematically by means of determinants. The vivid plots and dynamics analyses for double and triple-pole soliton solutions with the ZBCs as well as double and triple-pole interaction solutions with the NZBCs are exhibited in details. Compared with the most classical second-order flow Kaup-Newell system, we find the third-order dispersion and quintic nonlinear term of the Kaup-Newell system change the trajectory and velocity of solutions. Furthermore, the asymptotic states of the 1-double poles soliton solution and the 1-triple poles soliton solution are analyzed when t tends to infinity.

Dynamics of localized waves for the higher-order nonlinear Schrödinger equation with self-steepening and cubic–quintic nonlinear terms in optical fibers

Dynamics of localized waves for the higher-order nonlinear Schrödinger equation with self-steepening and cubic–quintic nonlinear terms in optical fibers

The optical soliton solutions of nonlinear Schrödinger equation with quintic non-Kerr nonlinear term

The purpose of this paper is to study the optical soliton solutions of nonlinear Schrödinger equation with quintic non-Kerr nonlinear term describing the nonlinear wave state of optical solitons, which is a noteworthy and important model in optical fiber communication. By using the polynomial complete discrimination system, the various optical soliton solutions of the nonlinear Schrödinger equation are obtained including the trigonometric function solution, the hyperbolic function solution and the Jacobian elliptic function solution. Finally, the graphs of some obtained solutions are drawn by setting values to the parameters.

A hybrid averaging and harmonic balance method for weakly nonlinear asymmetric resonators

Models for nonlinear vibrations commonly employ polynomial terms that arise from series expansions about an equilibrium point. The analysis of symmetric systems with cubic stiffness terms is very common, and the inclusion of asymmetric quadratic terms is known to modify the effective cubic nonlinearity in weakly nonlinear systems. When using low (second, in this case)-order perturbation methods, the net effect in these cases is found to be a monotonic dependence of the free vibration frequency on the amplitude squared, with a single term that depends on the coefficients of the quadratic and cubic terms. However, in many applications, such a monotonic dependence is not observed, necessitating the use of techniques for strongly nonlinear systems, or the inclusion of higher-order terms and perturbation methods in weakly nonlinear formulations. In either case, the analysis involves very tedious and/or numerical approaches for determining the system response. In the present work, we propose a method that is a hybrid of the methods of averaging and harmonic balance, which provides, with relatively straightforward calculations, good approximations for the free and forced vibration response of weakly nonlinear asymmetric systems. For free vibration, it captures the correct amplitude–frequency dependence, including cases of non-monoticity. The method can also be used to determine the steady-state response of damped, harmonically driven vibrations, including information about stability. The method is described, and general results are obtained for an asymmetric system with up to quintic nonlinear terms. The results are applied to a numerical example and validated using simulations. This approach will be useful for analyzing a variety of system models with polynomial nonlinearities.

KAM Tori for a Two Dimensional Beam Equation with a Quintic Nonlinear Term and Quasi-periodic Forcing

KAM Tori for a Two Dimensional Beam Equation with a Quintic Nonlinear Term and Quasi-periodic Forcing

Vortex gap solitons in spin–orbit-coupled Bose–Einstein condensates with competing nonlinearities

The formation and dynamics of full vortex gap solitons (FVGSs) are investigated in two-component Bose–Einstein condensates with spin–orbit coupling (SOC), Zeeman splitting (ZS), and competing cubic and quintic nonlinear terms, while the usual kinetic energy is neglected, assuming that it is much smaller than the SOC and ZS terms. Unlike previous SOC system with the cubic-only attractive nonlinearity, in which solely semi-vortices may be stable, with the vorticity carried by a single component, the present system supports stable FVGS states, with the vorticity present in both components (such states are called here “full vortex solitons”, to stress the difference from the half-vortices). They populate the bandgap in the system’s linear spectrum. In the case of the cubic self-attraction and quintic repulsion, stable FVGSs with a positive effective mass exist near the top of the bandgap. On the contrary, the system with cubic self-repulsion and quintic attraction produces stable FVGSs with a negative mass near the bottom of the bandgap. Mobility and collisions of FVGSs with different topological charges are investigated too.

Invariant Tori for a Two-Dimensional Completely Resonant Beam Equation with a Quintic Nonlinear Term

This paper focuses on a two-dimensional completely resonant beam equation with a quintic nonlinear term. This means studying u t t + Δ 2 u + ε f u = 0 , x ∈ T 2 , t ∈ ℝ , under periodic boundary conditions, where ε is a small positive parameter and f u is a real analytic odd function of the form f u = f 5 u 5 + ∑ i ^ ≥ 3 f 2 i ^ + 1 u 2 i ∧ + 1 , f 5 ≠ 0 . It is proved that the equation admits small-amplitude, Whitney smooth, linearly stable quasiperiodic solutions on the phase-flow invariant subspace ℤ † 2 = r = r 1 , r 2 , r 1 ∈ 4 ℤ − 1 , r 2 ∈ 4 ℤ . Firstly, the corresponding Hamiltonian system of the equation is transformed into an angle-dependent block-diagonal normal form by using symplectic transformation, which can be achieved by selecting the appropriate tangential position. Finally, the existence of a class of invariant tori is proved, which implies the existence of quasiperiodic solutions for most values of frequency vector by an abstract KAM (Kolmogorov-Arnold-Moser) theorem for infinite dimensional Hamiltonian systems.

On vortex and dark solitons in the cubic–quintic nonlinear Schrödinger equation

We study topologically charged propagation-invariant eigenstates of the 1+2-dimensional Schrödinger equation with a cubic (focusing)–quintic (defocusing) nonlinear term. First, we revisit the self-trapped vortex soliton solutions. Using a variational ansatz that allows us to describe the solutions as a liquid with a surface tension, we derive a simple formula relating the inner and outer radii of the bright vortex ring. Then, using numerical and variational techniques, we analyse dark soliton solutions for which the wave function density asymptotes to a non-vanishing value. We find an eigenvalue cutoff for the propagation constant that depends on the topological charge l. The variational profile provides simple and very accurate results for l≥2. We also study the azimuthal stability of the eigenstates by a linear analysis finding that they are stable for all values of the propagation constant, at least for small l.

Controllable transmission of Airy pulses in nonlinear dissipative system

The propagation characteristics of truncated Airy pulses have been numerically investigated in complex cubic-quintic Ginzburg-Landau equation and some interesting nonlinear phenomena are observed. The results show that the transmission trajectory and amplitude of the Airy pulses can be controlled by changing the gain dispersion. Due to the complex interaction between the side lobes, the double Airy pulses evolve into single soliton, double solitons and triple solitons, respectively. The interaction between linear loss and nonlinear gain has a significant effect on the propagation dynamics of Airy pulses, which can obtain stable bound state solitons or pulsating-like solitons. In addition, the period, structure and number of solitons can be controlled by properly choosing the parameters of the quintic nonlinear correction term.

Free and forced nonlinear vibrations of Bi-Directional functionally graded Euler–Bernoulli porous beams

In this study, the free and forced vibrations of a bi-directional functionally graded porous beam are investigated. The governing equations of motion are derived by Hamilton’s principle, and the reduced temporal equation of motion with cubic and quintic nonlinear terms is obtained using the Galerkin approach. Analytical solutions for the nonlinear natural frequencies in addition to the primary resonance response curves are established by the method of multiple scales. The effects of the axial and transverse functionally graded indexes, initial amplitude, porosity parameter, and the elastic and mass density ratios on the nonlinear frequencies and the forced responses are examined.

Bursting oscillations and bifurcation analysis for a Filippov system with a quintic nonlinear term

Bursting oscillations and bifurcation analysis for a Filippov system with a quintic nonlinear term

Role of the quintic nonlinear refractive term in the stability of dissipative solitons of the complex Ginzburg–Landau equation

We revisit the role of the quintic terms of the complex cubic-quintic Ginzburg–Landau equation in the generation of stable dissipative solitons. Using direct numerical simulations and a qualitative analysis, we show that the presence of one of the two quintic terms is a sine qua non. However, this term is not necessarily the quintic gain saturation term as had been demonstrated by Moores [Opt. Commun. 96, 65 (1993)OPCOB80030-401810.1016/0030-4018(93)90524-9] but can be the higher-order (quintic) nonlinear refraction term. We prove that by numerically solving this equation, and we perform a qualitative analysis that shows that the negative soliton chirp, anomalous dispersion, and spectral filtering are the physical effects responsible for gain saturation in this case.

Soliton interaction of a generalized nonlinear Schrödinger equation in an optical fiber

Under investigation in this work is a generalized nonlinear Schrödinger (NLS) equation with high-order dispersion and quintic nonlinear terms, which can describe a subpicosecond pulse propagation in optical fibers. By developing Hirota method, the bilinear form and analytical one- and two-soliton solutions are derived. Based on the analytical solutions, we give the corresponding soliton patterns to analyse dynamics of the pulse solitons. It shows that high order dispersion term may change the periodicity of propagation. The interaction of two pulse solitons is an elastic collision. By choosing suitable parameter values, we can obtain the two parallel solitons. It can be applied to improve transmission quality and capacity of information in optical fiber.

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.