Quintic Terms Research Articles

From deterministic to stochastic: limits of extracting bifurcation diagrams for noisy bistable oscillators with the control-based continuation method

AbstractNoise limits the information that can be experimentally extracted from dynamical systems. In this study, we review the Control-based Continuation (CBC) approach, which is commonly used for experimental characterisation of nonlinear systems with coexisting stable and unstable steady states. The CBC technique, however, uses a deterministic framework, whereas in practice, almost all measurements are subject to some level of random perturbation, and the underlying dynamical system is inherently noisy. In order to discover what the CBC is capable of extracting from inherently noisy experiments, we study the Hopf normal form with quintic terms with additive noise. The bifurcation diagram of the deterministic core of this system is well-known, therefore the discrepancies introduced by noise can be easily assessed. First, we utilise the Step-Matrix Multiplication based Path Integral (SMM-PI) method to approximate the system’s steady state probability density function (PDF) for different intensity noise perturbations. We associate the local extrema of the resulting PDFs with limit cycles, and compare the resulting bifurcation diagram to those captured by CBC. We show that CBC estimates the bifurcation diagram of the noisy system well for noise intensities varying from small to moderate, and in practice, the amplitudes provided by CBC may be accepted as a ’best guess’ proxy for the vibration amplitudes characteristic to the near periodic solutions in a wide range of experiments.

Nonlinear incompressible shear wave models in hyperelasticity and viscoelasticity frameworks, with applications to Love waves

General equations describing shear displacements in incompressible hyperelastic materials, holding for an arbitrary form of strain energy density function, are presented and applied to the description of nonlinear Love-type waves propagating on an interface between materials with different mechanical properties. The model is valid for a broad class of hyper-viscoelastic materials. For the Murnaghan constitutive model, shear wave equations contain cubic and quintic differential polynomial terms, including viscoelasticity contributions in terms of dispersion terms that include mixed derivatives uxxt of the material displacement. Full (2+1)-dimensional numerical simulations of waves propagating in the bulk of a two-layered solid are undertaken and analysed with respect to the source position and mechanical properties of the layers. Interfacial nonlinear Love waves and free upper surface shear waves are tracked; it is demonstrated that in the fully nonlinear case, the variable wave speed of interface and surface waves generally satisfies the linear Love wave existence condition c1<v<c2, while tending to the larger material wave speed c1 or c2 for large times.

The unified transformation approach to higher-order Gerdjikov-Ivanov model and Riemann-Hilbert problem

In this paper, a higher-order Gerdjikov-Ivanov (HOGI) model with cubic dispersion and quintic nonlinearity terms will be researched, which is a 3-order flow model of the well known GI system. The initial-boundary value problems of the HOGI model on the half-line will be given by the unified transformation approach. The solutions of the HOGI model will be expressed by forms of the solutions of a matrix Riemann-Hilbert problem (RHP). The relevant jump matrices of the RHP will be explicitly found by the two scattering matrices s(ϵ) and S(ϵ). Finally, the so-called global relation will be given.

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.

Motion dynamics of two-dimensional fundamental and vortex solitons in the fractional medium with the cubic-quintic nonlinearity

Motion dynamics of two-dimensional fundamental and vortex solitons in the fractional medium with the cubic-quintic nonlinearity

Lie symmetry analysis of cubic–quartic optical solitons having cubic–quintic–septic–nonic form of self-phase modulation structure

AbstractThe present study examines optical solitons characterized by cubic–quartic dynamics and featuring a self-phase modulation structure encompassing cubic, quintic, septal, and nonic terms. Soliton solutions are obtained through Lie symmetry analysis, followed by integration of the resulting ordinary differential equations using Kudryashov’s auxiliary equation method and a hyperbolic function approach. A comprehensive range of optical soliton solutions has been recovered, alongside the revelation of their criteria for existence.

Linear dynamics of a thick liquid layer subjected to an oblique temperature gradient

The present study aims to demonstrate the application of the oblique temperature gradient (OTG) to manipulate the buoyancy instabilities and resulting mixing. To accomplish this, we consider a model consisting of a liquid layer supported by a perfectly conducting bottom wall from below and the upper surface is exposed to ambient inert gas in a shallow container. The stability analysis employs the pseudospectral method and perturbation energy budget approach. The imposed OTG consists of a negative vertical temperature gradient (VTG), which leads to unstable density stratification, and a negative horizontal temperature gradient (HTG) component responsible for the Hadley circulation, which can be utilised to control instabilities. Without the HTG, a Rayleigh–Bénard mode of instability exists due to the imposed VTG termed as a VTG mode. The imposed HTG induces a linear VTG term and a quintic polynomial VTG term in the bottom wall-normal coordinate $y$ . The induced linear VTG term reinforces the imposed VTG, while the induced quintic polynomial VTG term opposes the imposed VTG. For the vanishing Biot number ( $Bi$ ), the induced quintic VTG term stabilises the VTG mode for the Prandtl number, $Pr=7$ . However, for $Pr=0.01$ , the Reynolds stresses associated with the Hadley circulation destabilise the VTG mode. For non-zero $Bi$ , new streamwise and spanwise instability modes arise for $Pr=7$ due to the induced linear VTG term. Physical arguments show that the unstable density stratification near the gas–liquid interface caused by the synergistic effect of the imposed VTG and the induced linear VTG term lead to the existence of the predicted new modes. The present study thus shows that the strength of the HTG can be utilised to control the instability modes and critical parameters, thereby demonstrating the utility of the OTG in manipulating the buoyancy instabilities.

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

Formation and propagation of fundamental and vortex soliton families in 1D and 2D dual-Lévy-index fractional nonlinear Schrödinger equations with cubic–quintic nonlinearity

We address effects of interplay of two different fractional-diffraction terms, corresponding to different Lé vy indices (LIs), in the framework of nonlinear Schrödinger equation (NLSE) with cubic–quintic nonlinearity (dual-LI fractional NLSE ), in one- and two-dimensional (1D and 2D) settings. The critical (in 1D) and supercritical (in 2D) wave collapses are suppressed in the presence of a defocusing quintic term, making it possible to produce stable localized modes, including fundamental and vortical ones. In 2D, families of fundamental and vortex solitons, with topological charges S=0,1,2,3, are produced in a numerical form and by dint of the variational approximation (VA). In particular, the threshold power necessary for the existence of 2D fundamental solitons is predicted by the VA, being close to the numerical results. In the case of fractional-diffraction terms with unequal LIs acting in two transverse directions, anisotropic 2D fundamental and vortex solitons are constructed. Stability of the solitons is investigated for small perturbations governed by the linearized equations, and results are corroborated by direct simulations of the perturbed evolution. Those vortex solitons which are unstable are split by azimuthal perturbations into fragments, whose number is determined by the shape of the perturbation eigenmode with the largest growth rate. These results extend the concept of the 2D fractional diffraction and solitons to media with the anisotropic fractionality.

Exact solutions to the fractional complex Ginzburg-Landau equation with cubic-quintic and Kerr law nonlinearities

The fractional complex cubic-quintic Ginzburg-Landau equation (FCCQGLE) with variable coefficients is a propagation model of optical pulses in optical fibers, the quintic term contained in it not only explains the physical significances that are not found in the existing models, but also has more abundant dynamic characteristics compared with lower dimensional systems. In this paper, the exact solutions of this equation are obtained via the appropriate transformations methods, which also are divided into different kinds. More precisely, we acquire the solitary, soliton and elliptic waves solutions by using the improved unified method, the bright and dark solitons solutions by using the improved F-expansion method, as well as the traveling wave solutions through the improved (G′/G2) -expansion and Bernoulli sub-equation function methods. Furthermore, we draw the 3D, 2D, density and contour plots to understand the propagation forms and the changes of amplitudes, frequencies and shapes of the solutions for the FCCQGLE with variable coefficients. The last thing worth mentioning is that the improved unified method here extends the degree of the polynomial from n to -n in the hypothetical solutions, which is not appeared before.

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.

Nonlinear WEC modeling using Sparse Identification of Nonlinear Dynamics (SINDy)

Modeling oscillating surge wave energy converter (OSWEC) systems to accurately predict their behavior has been a notoriously difficult challenge for the wave energy field. This is particularly challenging in realistic sea states where nonlinear WEC dynamics are common due to complex fluid-structure interaction, breaking waves, and other phenomena. Common modeling techniques for OSWECs include using potential flow theory to linearize the governing equations and ease computations, or using CFD to solve the full Navier-Stokes equations coupled with rigid body motion. However, both of these options have significant limitations. Potential flow theory breaks down in realistic sea conditions where nonlinear WEC dynamics are present, and CFD is often too computationally expensive for many applications such as real-time state prediction and optimal control, two areas of active research in the wave energy field. In particular, OSWEC dynamics are dominated by diffractive and viscous forces, often making common assumptions and linearization approximations (including small-body approximations) unreasonable, and CFD computationally intractable.&#x0D; To bridge this gap in modeling methods, we propose using Sparse Identification of Nonlinear Dynamics (SINDy) to build nonlinear reduced-order models (ROMs) that describe OSWEC behavior in response to large-amplitude regular waves. SINDy is an equation-free, data-driven algorithm that identifies dominant nonlinear functions present in system state dynamics using a library of nonlinear functions created from time series measurement data. The result is an ordinary differential equation (ODE) in time that can be solved from an initial condition to model and predict time behavior of the states. SINDy is parsimonious, meaning it uses a sparsity-promoting hyperparameter with the goal of only including the minimum number of terms to capture dominant dynamics, resulting in interpretable and generalizable results that are not overfit to the data. Using the discovered ROMs and integrating in time, not only can SINDy provide time series models and future state predictions of OSWEC dynamics, it can also give insights into which variables are critical in describing the underlying dynamics of the state. &#x0D; In this study, we use SINDy to describe the nonlinear dynamics of a lab-scale OSWEC in a wave tank subjected to large-amplitude regular waves. We use nonlinear simulation data to generate kinematic, force, and torque data and use it as input to SINDy to identify ODEs that describe the measurement variables in time. We then integrate the ODEs to recreate the time series as well as predict future system behavior. We directly compare the resulting time series to the original data input to assess the accuracy of the SINDy model. We also interpret the dominant terms in the ODEs to gain insight on underlying mechanisms of the observed nonlinearity.&#x0D; Early results show SINDy is a promising tool for modeling nonlinear OSWEC dynamics. We are able to build ROMs for variables such as angular kinematics and the moment about the hinge that generate an accurate recreation of data measurements. We found strong dominance in cubic and quintic terms of the ROMs, suggesting higher-order nonlinearities in the system dynamics. These findings inspire future work in identifying underlying mechanisms driving nonlinearity.

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

On the higher-order Heisenberg spin system with quintic terms in self-consistent potential

The integrable higher-order Heisenberg spin system with quintic terms in self-consistent potential (HHSQ) is an extension of Heisenberg ferromagnetic equations. In this paper, we calculate the Lax expression of the HHSQ system, then the motioning curves in [Formula: see text] dimension of Heisenberg ferromagnetic equations have been discussed. The Darboux transformation of the HHSQ system is constructed and a set of soliton solutions will be demonstrated by Darboux transformation.

Rogue wave solutions over double-periodic wave background of a fifth-order nonlinear Schrödinger equation with quintic terms

Studies on rogue waves (RWs) over constant and periodic wave background are of contemporary interest in several areas of physics. These RWs arise not only on such backgrounds but also appear on double-periodic wave background. In this paper, we derive RW solutions over a double-periodic wave background of a fifth-order nonlinear Schrödinger equation. We create the double-periodic wave background with elliptic functions (in combinations of cn, sn and dn) as seed solutions in the first iteration of Darboux transformation. We then calculate the growth rate of instability for these double-periodic solutions under different values of elliptic modulus parameter. On the top of these background waves, we generate RWs with the help of second independent solution to the eigenvalue problem. Further, we analyze the differences that occur in the appearance of RWs with reference to the lower-order and higher-order dispersions terms. In addition to the above, we examine the derived solutions in detail for certain system and elliptic modulus parameter values and highlight some interesting features that we obtain from our studies.

Group classification and exact solutions of fractional differential equation with quintic non-Kerr nonlinearity term

Group classification and exact solutions of fractional differential equation with quintic non-Kerr nonlinearity term