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In this article, the gamma function method, for the first time ever, is used to solve the nonlinear cubic-quintic Duffing oscillators. The nonlinear cubic-quintic Duffing oscillators with and without the damped and quadratic terms are considered respectively. By the gamma function method, it only needs one-step to get the approximate solution. The comparisons with the existing solutions reveal that the proposed method is simple but effective in solving the small amplitude oscillation.

To achieve $$G^3$$ Hermite interpolation with a lower degree curve, this paper studies planar $$G^3$$ Hermite interpolation using a quintic Bezier curve. First, the first and second derivatives of the quintic Bezier curve satisfying $$G^2$$ condition are constructed according to the interpolation conditions. Four parameters are introduced into the construction. Two of them are set as free design parameters, which represent the tangent vector module length of the quintic Bezier curve at the two endpoints, and the other two parameters are derived from $$G^3$$ condition. Then, to match $$G^3$$ condition, it is necessary to ensure that the first derivative of curvature with respect to arc length is equal. Nevertheless, the direct calculation of the derivative of curvature involves the calculation of square root. Alternatively, an equivalent condition is derived by investigating the first derivative of curvature square. Based on this condition, the two parameters can be computed as the solutions of linear systems. Finally, the control points of the quintic Bezier curve are obtained. Several comparative examples are provided to demonstrate the effectiveness of the proposed method. A variety of complex shape curves can be obtained by adjusting the two free design parameters. Applications to shape design are also shown.

In this manuscript, a novel modification of quintic B-spline differential quadrature method is given. By this modification, the obtained algebraic system do not have any ghost points. Therefore, the equation system is solvable and doesn't need any additional equation. Another advantage of this modification is the fact that the obtained numerical results are better than classical methods. Finite difference method is used as another component of the algorithm to obtain numerical solution of Korteweg-de Vries (KdV) equation which is the important model of nonlinear phenomena. Nine effective experiments namely single soliton, double solitons, interaction of two solitons, splitting to the solitons, interaction of three-, four-, and five solitons, evolution of solitons via Maxwellian initial condition and train of solitons are given with comparison of error norms and invariants. All of the given results with earlier studies show that the present scheme has improved classical common methods. Rate of convergence for single soliton and double solitons problems are reported.

This paper deals with three degrees of freedom Hamiltonian system in the case that the frequencies of the linearization are in 1:2:3 resonance. We obtain the truncated second order normal form expressed with the basic invariants. Also, we identify the coefficients of the terms that remain in normalization procedure. We concentrate on determining the relative equilibria with or without considering a perturbation in frequencies. By a homogeneous structure, all relative equilibria are computed and the linear stability of isolated relative equilibria is investigated. Especially we find some parametric manifold of relative equilibria and a quintic polynomial which determines relative equilibria.

In this work, we investigate and apply higher-order Hamiltonian approach (HA) as one of the novelty techniques to find out the approximate analytical solution for vibrating double-sided quintic nonlinear nano-torsional actuator. Periodic solutions are analytically verified, and consequently, the relationship between the initial amplitude and the natural frequency are obtained in a novel analytical way. The HA is then extended to the second-order to find more accurate results. To show the accuracy and applicability of the technique, the approximated results are compared with the homotopy perturbation method and numerical solution. According to the numerical results, it is highly remarkable that the second-order approximate solutions produce better than previously existing results and almost similar in comparing with the numerical solutions.

We prove that solutions to the quintic semilinear wave equation with variable coefficients in $\mathbb R^{1+3}$ scatter to a solution to the corresponding linear wave equation. The coefficients are small and decay as $|x|\to\infty$, but are allowed to be time dependent. The proof uses local energy decay estimates to establish the decay of the $L^6$ norm of the solution as $t\to\infty$., Comment: 21 pages

The paper gives a comprehensive study of infinite-energy solutions and their long-time behavior for semi-linear weakly damped wave equations in R 3 \mathbb {R}^3 with quintic nonlinearities. This study includes global well-posedness of the so-called Shatah-Struwe solutions, their dissipativity, the existence of a locally compact global attractors (in the uniformly local phase spaces) and their extra regularity.

The shape curve of continuous variable crown (CVC) roll applied in plate rolling process is usually adopted as cubic curves or quintic curves, but the coupling between quadratic crown and quartic crown of conventional CVC roll shape curve could not work well and failed to meet the product quality requirements in plate rolling. To get better control effect on the composite wave shape, a sextic CVC roll shape curve was proposed. The theoretical calculation formula of the roll shape parameters was deduced by using the decomposition and superposition principle. Based on the actual production data of plate mill, sextic CVC work roll parameters were determined through the analysis of the convexity control ability of the curve. The application results showed that a sextic CVC roll shape curve can greatly improve the control ability of plate crown. The control ability of quadratic crown improved by 5% and that of quartic crown improved by 20%, respectively; further, the peak contact pressure of quintic CVC roll decreased from 2076 to 1744 MPa, and the difference between the maximum and the minimum of roll pressure was 184 MPa. The control ability of plate crown was controlled from − 67 to 63 μm with plate width equaling to 2400 mm. The deviation of measured value of plate crown and calculated value of plate crown was less than ± 4%.

A subset of traveling wave solutions of the quintic complex Ginzburg-Landau equation (QCGLE) is presented in compact form. The approach consists of the following parts: 1) Reduction of the QCGLE to a system of two ordinary differential equations (ODEs) by a traveling wave ansatz; 2) Solution of the system for two (ad hoc) cases relating phase and amplitude; 3) Presentation of the solution for both cases in compact form; 4) Presentation of constraints for bounded and for singular positive solutions by analysing the analytical properties of the solution by means of a phase diagram approach. The results are exemplified numerically.

We consider the nonlinear Schrodinger equation in three space dimensions with combined focusing cubic and defocusing quintic nonlinearity. This problem was considered previously by Killip et al. [A...

In this work, we use the inverse scattering transform method to consider the focusing Kundu–Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the analytical, symmetric, asymptotic properties of eigenfunctions, the inverse problem is solved via a matrix Riemann–Hilbert problem (RHP). The general multi-pole solutions are given in terms of the solution of the associated RHP. And the formula of the N simple-pole soliton solutions are obtained, too. We show that the Peregrine’s rational solution can be viewed as some appropriate limit of the simple-pole soliton solutions at branch point. Furthermore, by taking some other proper limits, the two and three simple-pole soliton solutions can yield to the double- and triple-pole solutions for focusing KE equation with NZBG. The effect of the parameter $$\beta $$ , characterizing the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effect, and some typical collisions between solutions of focusing KE equation are graphically displayed.

We provide the sufficient conditions for the simultaneous existence of centres for two families of planar double-reversible quintic systems. Computing the focal values and using crossed resultants ...

The nonlinear oscillators describing by differential equations of the form x 00 = −ax + cx5 and x 00 = −ax + bx3 + cx5 are studied. Multiplicity results for both types of equations, given with the Neumann boundary conditions are obtained. It is shown that the number of solutions depend on the coefficient a only. The exact estimates of the number of solutions are obtained. Practical issues, such as the representation of solutions in terms of Jacobian elliptic functions and calculation of the initial values for solutions of boundary value problems, are considered also. The illustrative examples are provided. Outlines of future research conclude the article.

Nonlinear transversely vibrating beams, including a uniform beam carrying a lumped mass and a transversely vibrating quintic nonlinear beam, are considered in this paper. Firstly, using trigonometric function, analytical solutions to their Cauchy initial problems are constructed in parametric and closed-form. Secondly, it is found that one has the freedom to choose any periodic function to simulate the periodic vibration theoretically. As an example, we also construct parametric and closed-form solution expressed by Jacobi elliptic function. Thirdly, by comparing the two kinds of derived solutions, it is shown that the Jacobi elliptic function solution can degenerate to the corresponding trigonometric function solution when the modulus tends to zero. Comparison are also made between our derived Jacobi elliptic function solution and other’s exact solution, which indicates that the presented parametric solution method is more general.

The nonlinear dispersive cubic-quintic Schrdinger equation of higher order studies in this paper by employing two analytical schemes. The exp ( − ϕ ( ξ ) ) -expansion and extended simple equation methods are used and exact solutions are attained in several kinds in which some are novel. According to our best knowledge, some solutions do not exist earlier. These solutions have key applications in engineering and applied physics. The attained results can be utilized to elucidate and know the physical nature of waves spread in the dispersive optical medium. A few attractive shapes are also depicted for the interpretation physically of the achieved solutions. Several such types of other models occurring in physics and other fields of applied science can be solved by using this consistent, influential and effective technique.

We have studied numerically the influence of intrapulse Raman scattering (IRS) and self-steepening (SS) on the period-2 pulsating solution of the complex cubic-quintic Ginzburg–Landau equation. A cascade of transformations of the numeric solutions under the influence of SS is reported, which includes the existence of: period-1 solutions, chaotic solutions, period–doubling transformation leading to the appearance of pulsating solutions with many periods, then period–halved transformation, periodic in t and x solutions and, finally, uniformly translating fronts. We have shown that by increasing the IRS parameter, the period-4 pulsating solution related to the SS can be successfully transformed into a period-2, period-1 pulsating solutions and, finally, into a stationary solution.

In this paper, a variable-coefficient cubic–quintic nonlinear Schrodinger equation involving five arbitrary real functions of space and time is analyzed from the point of view of symmetry analysis by using Lie’s invariance infinitesimal criterion. The infinitesimal generators of corresponding equivalence transformations are presented. The first-order differential invariants are constructed to identify when the equation can be mapped to a constant-coefficient cubic–quintic nonlinear Schrodinger equation. The constrained conditions on the variable coefficients we arrived here extend the cases discussed before and present more general results. Some brightlike and darklike solitary wave solutions for special potentials and cubic–quintic nonlinearities are obtained.

We present a B-spline based semi-analytical technique for solving 2D convection-diffusion-reaction equations. The main feature of the presented technique is to separate the satisfaction of the conditions on the boundary and the elliptic partial differential equation inside. To be more precise, we transform the original equation into the problem with homogeneous boundary conditions and seek the approximate solution as a sum of the modified B-spline tensor products which satisfy the homogeneous boundary conditions of the problem. The cubic and quintic B-spline are used in the framework of the method. The coefficients of linear combination are determined to satisfy the governing equation. Eight numerical examples have been studied to demonstrate the high effectivity of the presented technique in solving 2D convection-diffusion-reaction problems in single and multi-connected domains.

In this paper, we introduce the third-order flow equation of the Kaup–Newell (KN) system. We study this equation, and we obtain different types of solutions by using the Darboux transformation (DT) and the extended DT of the KN system, such as solitons, positons, breathers, and rogue waves. The extended DT is obtained by taking the degenerate eigenvalues $$ \lambda _{i} \rightarrow \lambda _{1} (i=3,5,7,\ldots ,2k-1)$$ and by performing the Taylor expansion near $$\lambda _{1}$$ of the determinants of DT. Some analytic expressions are explicitly given for the first-order solutions. We study the unique waveforms of both the first-order and higher-order rogue-wave solutions for special choices of parameters, and we find different types of such wave structures: fundamental pattern, triangular, modified-triangular, pentagram, ring, ring-triangular, and multi-ring wave patterns. We conclude that the third-order dispersion and quintic nonlinear term of the KN system modify both the trajectories and speeds of the solutions as compared with those corresponding to the second-order flow equation of the KN system.

This is a detailed study of damped quintic wave equations with non-regular and non-autonomous external forces which are measures in time. In the 3D case with periodic boundary conditions, uniform energy-to- Strichartz estimates are established for the solutions, the existence of uniform attractors in a weak or strong topology in the energy phase space is proved, and their additional regularity is studied along with the possibility of representing them as the union of all complete bounded trajectories. Bibliography: 45 titles.

In this paper, a complex cubic–quintic Ginzburg–Landau equation (CCQGLE) is investigated. Using the asymmetric method, the analytic one-soliton solution of the CCQGLE is obtained for the first time. Through analyzing the solutions obtained, the transmission of the soliton is controlled by changing the values of related parameters. Results of this paper contribute to obtain the analytic soliton solution of the higher-order CCQGLE.

We find analytically an exact soliton-like solution of cubic-quintic nonlinear Schrodinger equation (CQNLSE) with pure normal fourth-order dispersion (FOD). This exact solitary solution is “fixed-parameter” solitary wave, whose waveform is well described by hyperbolic secant function without oscillating tails. Those characteristics contribute importantly new information on solitons generation compared to the conventional pure-quartic soliton (PQS) in the absence of quintic nonlinearity. The numerical results verify that this exact solitary solution can preserve its shape. The formation of such solitary wave is a result of counterbalance between normal FOD as well as positive Kerr nonlinearity and negative quintic nonlinearity. The stability of such exact solitary wave is discussed as well. Although such exact solitary wave is linearly unstable in the presence of perturbation, our results disclose the role of quintic nonlinearity in the formation of high-energy PQS and its propagation, and also provide a way to control the generation of high-energy PQS laser pulse.

The fractional ( 3 + 1 ) -dimensional nonlinear Schrodinger equation with cubic–quintic–septic nonlinearities plays a significant role in the study of ultra-short pulses in highly nonlinear optical phenomena. The main purpose of this work is to determine the solution of ( 3 + 1 ) -dimensional nonlinear Schrodinger equation containing cubic–quintic–septic nonlinearities with conformal temporal operator. The solution of the considered problem is investigated using an adaptation of the residual power series method for the conformal fractional derivative. To illustrate the authenticity of the residual power series method to solve the nonlinear conformable Schrodinger equation with cubic–quintic–septic nonlinearities, three test applications are considered subject to different initial conditions. The variations of wave solutions of the applications corresponding to changes in the conformal derivative are depicted through graphical illustrations. The numerical comparisons confirm the accuracy of the presented results for the conformal ( 3 + 1 ) -dimensional nonlinear Schrodinger equation. The obtained results indicate the accuracy, suitability and competency of the residual power series method to examine other nonlinear conformable fractional differential equations arising in optics and other areas of physics.

In this work, we construct a class of $$C^1$$ quintic interpolation basis functions over type-I triangulations with two shape parameters. The given interpolation basis functions have the properties of compact support, non-negativity and partition of unity. Based on the new interpolation basis functions, a kind of $$C^1$$ quintic triangular interpolation spline surfaces with two shape parameters is proposed. The resulting spline surfaces interpolate control points associated with barycentric parametric triples arranged in type-1 triangulations directly, without using any prescribed derivatives at any point of the domain. The local Bernstein–Bezier representation of the triangular interpolation spline surfaces is developed. In addition, the effects of the two shape parameters on generating the Bezier control points of the local surface patch are illustrated. Moreover, by blending the new $$C^1$$ quintic interpolation basis functions with linear parametrized basis surfaces, a kind of $$C^1$$ Overhauser-type interpolation spline surfaces with two shape parameters is also constructed.

The trigonometric quintic B-spline scheme is used in this research paper to research Zakharov’s (ZK) nonlinear dimensional equation’s numerical solution. The ZK model’s solutions explain the relationship between the high-frequency Langmuir and the low-frequency ion-acoustic waves with many applications in optical fiber, coastal engineering, and fluid mechanics of electromagnetic waves, plasma physics, and signal processing. Three recent computational schemes (the expanded [Formula: see text]-expansion method, generalized Kudryashov method, and modified Khater method) have recently been used to investigate this model’s moving wave solution. Many innovative solutions have been established in this paper to determine the original and boundary conditions that allow numerous numerical schemes to be implemented. Here, the trigonometric quintic B-spline method is used to analyze the precision of the collected analytical solutions. To illustrate the precision of the numerical and computational solutions, distinct drawings are depicted.

In this work, the variable-coefficients complex cubic–quintic Ginzburg–Landau equation (CCQGLE) influenced by higher-order effects and nonlinear gain is considered. Based on the asymmetric method, analytic one-soliton solution for the variable-coefficients CCQGLE is constructed for the first time. In addition, with some certain conditions, the periodic wave and dromion-like structures are derived. The results obtained may be helpful in understanding the solitons amplification and solitons management in optical fiber.

In this paper we provide a new method to study global dynamics of planar quasi-homogeneous differential systems. We first prove that all planar quasi-homogeneous polynomial differential systems can be translated into homogeneous differential systems and show that all quintic quasi-homogeneous but non-homogeneous systems can be reduced to four homogeneous ones. Then we present some properties of homogeneous systems, which can be used to discuss the dynamics of quasi-homogeneous systems. Finally we characterize the global topological phase portraits of quintic quasi-homogeneous but non-homogeneous differential systems.

Railway vehicles suffer from hunting motion, even when traveling below the critical speed obtained by linear analysis, due to the nonlinear characteristics of the wheel system. Nonlinear characteristics in Hopf bifurcations can be characterized as subcritical or supercritical, depending on whether the cubic nonlinearity is softening or hardening, respectively. In a system with softening cubic nonlinearity, third-order nonlinear analysis cannot detect nontrivial stable steady-state oscillations because they are affected by quintic nonlinearity. Therefore, in such a system, it is necessary to apply fifth-order nonlinear analysis to a system model in which quintic nonlinearity is taken into account. In this study, we investigated the cubic and quintic nonlinear phenomena in hunting motion with a roller rig that is widely used for hunting motion research. Previous experimental studies using a roller rig were restricted to the linear stability and the cubic nonlinear stability. We clarified that roller rig experiments can observe the hysteresis phenomenon and the existence of subcritical Hopf and saddle-node bifurcations, indicating that not only the cubic but also the quintic nonlinearity of the wheel system plays an important role. In addition, we obtained the normal form governing the nonlinear dynamics. We developed an experimental identification method to obtain the coefficients of the normal form. The validity of our method was confirmed by comparing the bifurcation diagrams obtained from the experimental time history and the normal form whose coefficients were experimentally identified using the proposed method.

A “transparent point” is a particular value of a governing parameter in a nontranslationally invariant system that makes the system “almost” translationally invariant. This concept was introduced recently in the context of the discrete nonlinear Schrodinger (DNLS) equation with saturable nonlinearity — it was discovered that a tuning of the lattice spacing parameter h in this model affects the soliton mobility. In this paper, we study the DNLS equation with competing cubic–quintic nonlinearity that also admits the transparent points with respect to the lattice spacing parameter h. We give a geometrical interpretation of the transparent points in terms of dynamical system theory and present a simple asymptotical formula for them at h → 0. Although the derivation of this formula is heuristic and nonrigorous, it gives the values of transparent points with remarkable accuracy even for quite large values of h.

We report on the existence and stability of Gaussian-type soliton in the nonlinear Schrodinger (NLS) equation with interplay of cubic–quintic nonlinearity, fourth-order diffraction (FOD) and novel quartic anharmonic parity-time ( $$\mathcal {PT}$$ )-symmetric Gaussian potential. We study numerically the impact of the FOD coefficient on the regions of unbroken/broken linear $$\mathcal {PT}$$ -symmetric phases. In the nonlinear domain, we derive exact soliton solutions of the one-dimensional and two-dimensional cubic–quintic NLS equation with $$\mathcal {PT}$$ -symmetric Gaussian potential and FOD coefficients. Moreover, the stability of the constructed soliton solution is investigated. The results of linear stability analysis are validated by comparison with numerical simulations. Furthermore, we also show that the relative strength of the FOD coefficient influences the direction of the power flow.

This paper successfully applied F-expansion algorithm to obtain highly dispersive optical solitons with cubic-quintic-septic nonlinearity. Bright, dark and singular solitons and their combinations thereof are listed. Their respective existence criteria are also indicated. Additional solutions such as periodic singular solutions as well as results in terms of Weierstrass elliptic function are also revealed from the scheme.

The extended Jacobi's elliptic function scheme is implemented to retrieve bright, dark and singular highly dispersive optical solitons that is studied in cubic–quintic–septic nonlinear medium. These solitons evolve from Jacobi's elliptic functions when limiting value of the modulus of ellipticity approaches unity.

Under investigation in this paper is a quintic time-dependent coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or a medium with the negative refractive index. A gauge transformation is found to obtain the equivalent form of the equation. With respect to the wave envelope for the free water surface displacement or envelope of the electric field, Painleve integrable condition, different from that in the existing literature, is derived, with which the bilinear forms and N-soliton solutions are constructed. Asymptotic analysis illustrates that the interactions between the bright and bound solitons as well as between the bright solitons and Kuznetsov–Ma breathers are elastic with certain conditions, while some other interactions are inelastic under other conditions. Propagation paths and velocities for the solitons are both affected by the dispersion coefficient function when the relations among the coefficients are linear, or affected by the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions when the relations among the coefficients are nonlinear. Under different conditions, bell-shaped solitons can evolve into the bound solitons or Kuznetsov–Ma breathers, respectively. Interactions between the bright and parabolic (or hyperbolic) solitons are related to the dispersion coefficient, self-steepening coefficient and cubic nonlinearity functions. Compression effect on the propagation paths of the solitons, caused by the dispersion coefficient, is observed.

In this article, we study the maximum number of limit cycles for a discontinuous quintic differential system. Using the first-order averaging method, we explain how limit cycles can bifurcate from the period annulus around the center of the considered system when it is perturbed inside a class of discontinuous quintic polynomial differential systems. Our results show that the lower bound and the upper bound of the number of limit cycles, 8 and 10 respectively, that can bifurcate from the period annulus around the center.

In this paper, we mainly study the number of limit cycles for a quintic Lienard system under polynomial perturbations. In some cases, we give new estimations for the lower bound of the maximal number of limit cycles.

In this paper, monotonicity-preserving interpolation is generalized to direction-consistent interpolation. The conditions for constructing direction-consistent tangent vectors are given. The conditions on the tangent vectors are also obtained such that the piecewise cubic Hermite interpolation curves are tangent direction-consistent with the direction of data polygon. Based on geometric insights, the balanced tangent vectors are presented and proved to be direction-consistent tangent vectors. With the balanced tangent vectors, the generated cubic Hermite interpolation curves are tangent direction-consistent, and the generated quintic Hermite interpolation curves are also tangent direction-consistent provided that we use the accumulative chord length parametrization. Some graphic examples are given to show that the generated interpolation curves preserve satisfactorily the shape of the given data control polygon.

We numerically analyzed the supratransmission phenomenon in the discrete nonlinear Schrodinger equation with the cubic–quintic nonlinearity. It has been reported that the homoclinic nonlinear band-gap threshold matches very well with the model. In the case of the cooperation between the nonlinearities (self-focusing cubic and quintic terms), the train of discrete band-gap waves overcomes the potential barrier of the first sites before merging or rebounding. In the case of competing self-focusing cubic and defocusing quintic nonlinearities, it is found that the lattice induces the generation of the train of dark solitons carried by a traveling kink and the traveling kink for chosen driving amplitude.

The differential geometric method of the center manifold theorem is applied to the magnetic-Lorenz model of ferrofluid convection in an electrically non-conducting ferrofluid. The analytically intractable tri-modal nonlinear autonomous system (magnetic-Lorenz model) is reduced to an analytically tractable uni-modal cubic–quintic Ginzburg–Landau equation. The inadequacy of the cubic Ginzburg–Landau equation and the need for the cubic–quintic one is shown in the paper. The heat transport is quantified using the solution of the cubic–quintic equation and the effect of ferrofluid parameters on it is demonstrated. The stable and unstable regions in the conductive regime and the conductive-convective regime is depicted using a bifurcation diagram. The noticeable discrepancy between the results of the two models is highlighted and the quintic non-linearity effects are delineated.

The dynamic response and bifurcations of high-dimensional systems endowed with hysteretic restoring forces in all degrees of freedom are investigated. Two types of hysteresis models are considered, namely the Bouc–Wen model and a differential version of the so-called exponential model of hysteresis. The numerical technique tailored for tackling high-dimensional hysteretic systems is based on an enhanced pathfollowing approach based on the Poincaré map. In particular, a five-dof mass-spring-damper-like system, with each rheological element described by the Bouc–Wen or the exponential model of hysteresis enriched by cubic and quintic nonlinear elastic terms, is investigated and a rich variety of nonlinear responses and bifurcations is found and discussed.

Under investigation in this work is a generalized nonlinear Schrodinger (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.

Free damped vibrations of a hinged–hinged Euler–Bernoulli beam subject to a constant axial force at its free end is investigated. The quintic nonlinear equation of motion is derived from Hamilton’s principle and then solved using the optimal auxiliary function method (OAFM). Our proposed procedure is highly efficient and controls the convergence of the solutions, ensuring an excellent accuracy after the first iteration. Numerical values are also obtained in order to validate the analytical results.

The Peregrine breather is an exact, rational and localized solution of the nonlinear Schrodinger equation, and is commonly employed as a model for rogue waves in physical sciences. If the transverse variable is allowed to be complex by analytic continuation while the propagation variable remains real, the poles of the Peregrine breather travel down and up the imaginary axis in the complex plane. At the turning point of the pole trajectory, the real part of the complex variable coincides with the location of maximum displacement of the rogue wave in physical space. This feature is conjectured to hold for at least a few other members of the hierarchy of Schrodinger equations. In particular, evolution systems with coherent coupling or quintic (fifth order) nonlinearity will be studied. Analytical and numerical results confirm the validity of this conjecture for the first and second order rogue waves.

A one-dimensional modified Nogochi nonlinear electric transmission network with dispersive elements that consist of a large number of identical sections is theoretically studied in the present paper. The first-order nonautonomous rogue waves with quintic nonlinearity and nonlinear dispersion effects in this network are predicted and analyzed using the cubic-quintic nonlinear Schrödinger (CQ-NLS) equation with a cubic nonlinear derivative term. The results show that, in the semidiscrete limit, the voltage for the transmission network is described in some cases by the CQ-NLS equation with a derivative term that is derived employing the reductive perturbation technique. A one-parameter first-order rational solution of the derived CQ-NLS equation is presented and used to investigate analytically the dependency of the characteristics of the first-order rouge wave parameters on the electric transmission network under consideration. Our results show that when we change the quintic nonlinear and nonlinear dispersion parameter, the first-order nonautonomous rogue wave transforms into the bright-like soliton. Our results also reveal that the shape of the first-order nonautonomous rogue waves does not change when we tune the quintic nonlinear and nonlinear dispersion parameter, while the quintic nonlinear term and nonlinear dispersion effect affect the velocity of first-rogue waves and the evolution of their phase. We also show that the network parameters as well as the frequency of the carrier voltage signal can be used to manage the motion of the first-order nonautonomous rogue waves in the electrical network under consideration. Our results may help to control and manage rogue waves experimentally in electric networks.

Given two spatial C 1 PH spline curves, aim of this paper is to study the construction of a tensor–product spline surface which has the two curves as assigned boundaries and which in addition incorporates a single family of isoparametric PH spline curves. Such a construction is carried over in two steps. In the first step a bi–patch is determined in a ‘Coons–like’ way having as boundaries two quintic PH curves forming a single section of given spline curves, and two polynomial quartic curves. In the second step the bi–patches are put together to form a globally C 1 continuous surface. In order to determine the final shape of the resulting surface, some free parameters are set by minimizing suitable shape functionals. The method can be extended to general boundary curves by preliminary approximating them with quintic PH splines.

Both one-dimensional and two-dimensional localized mode families in parity-time (𝒫𝒯)-symmetric potentials with competing cubic-quintic nonlinearities and higher-order diffraction are reported. In particular, we investigate the role played by the competing nonlinearities and fourth-order diffraction parameter on the beam dynamics in the generalized 𝒫𝒯-symmetric Scarf potentials. The numerical fundamental soliton in competing nonlinearities (1-D double peaked solitons) can be found to be stable around the propagation parameter for exact soliton. A linear stability analysis corroborated by direct numerical simulation reveals that the regions of stability of these solutions can be controlled by tuning the values of the FOD parameters as well as by tuning the sign of the cubic and quintic nonlinearities. In particular, we have shown that the FOD parameter can be used to provide the restoration of the stability of the solitons.

Curve fitting is an important process. Curve fitting is the process of constructing a curve that is closest to data series. In this research, curve fitting using quintic trigonometric Bezier curve (QTBC) with two shape parameters will be applied. Shape parameters will act as shape enabler in order to make the curves more flexible. Different degrees of parametric continuity are applied accordingly to connect each segment of two-dimensional objects to produce a smooth curve fitting. The curve fitting is made easier due to the presence of two parameters, making QTBC as the suitable tool for curve fitting. To analyse the performance, it is applied to three various objects that exhibit different features.

In this paper, continuous curves using different levels of continuity are being evaluated and compared as the degree of Bezier curve increases. Unfortunately, the ordinary Bezier curve lacks flexibility in controlling the shape of the curve. Therefore, quintic trigonometric Bezier curve is introduced as an alternative to produce a flexible continuous curve. The smooth continuous curve will act as a curve generator into producing desired shapes. Additionally, by applying quintic trigonometric Bezier curve on an object, the curve generator will undergo a sweeping technique in order to form a surface.