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This paper presents a general method for the trajectory planning of the redundant planar manipulator. The mathematical relation between joint space and Cartesian space in a two-dimensional space is first derived. The joint classification is employed to obtain the solutions of corresponding joints. It divides joints into class I redundant joints, class II redundant joints, and nonredundant joints. The new application of knot points in the quintic B-spline curve is introduced to generate inverse solutions of class I redundant joints. Examples show that the number and distribution of knot points have a large effect on their solutions. Moreover, the particle swarm optimization algorithm is extended to generate solutions of class II redundant joints. It also optimizes the initial trajectories of joints and end-effector. Finally, solutions of nonredundant joints can be generated by the derived relation between the joint space and Cartesian space. The proposed methodology is confirmed by a case study. Under the same conditions, results show that the solution obtained by the extended method is not only better than that obtained by the particle swarm optimization algorithm but also closer to the global optimal solution.

In this study, a high accurate numerical solution of the Burgers’ equation is obtained. For this, the collocation method based on quintic B-spline functions for space discretization and the fourth-order single step method for time discretization are used. In order to see the efficiency of the algorithm, a test problem with an analytical solution is discussed and compared with the numerical solution obtained.

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.

In this article, the modified Khater method is utilized to extract new analytical solutions to the cubic-quintic nonlinear non-paraxial pulse propagation model in a planar waveguide with Kerr nonlinearity, fifth-order nonlinearity, and spatiality dispersion due to non-paraxial effects. As a result, a variety of some new exact solutions are observed. Moreover, the three-dimensional shapes are visualized with the aid of the latest scientific tools.

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.

Applying a procedure similar to that of E.S. Bring, by using a 4th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a De Moivre form (DMQ), so that it could be solved by radicals. The general solution by radicals of the De Moivre equations of any degree is presented. By the same procedure the BJSx (normal sextic) equation was taken to another one without the 2nd, 4th and 6th terms which was transformed into a cubic (solvable) equation. By applying a 6th degree Tschirnhaus transformation to the BJSp (normal septic) equation its binormal (without the 2nd, 3rd, 4th and 5th terms) form was obtained.

Applying a procedure similar to that of E.S. Bring, by using a 4th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a De Moivre form (DMQ), so that it could be solved by radicals. The general solution by radicals of the De Moivre equations of any degree is presented. By the same procedure the BJSx (normal sextic) equation was taken to another one without the 2nd, 4th and 6th terms which was transformed into a cubic (solvable) equation. By applying a 6th degree Tschirnhaus transformation to the BJSp (normal septic) equation its Bring-Jerrard binormal (without the 2nd, 3rd, 4th and 5th terms) form was obtained.

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.

An efficient numerical technique for the solution of the pulp washing model is proposed in this study. Two linear and one nonlinear model are explained with quintic Hermite collocation method. In this technique, quintic Hermite polynomials (C2 continuous) are used as a basis function and orthogonal collocation method is applied within each element of the partitioned domain. For accuracy and applicability of the method, a comparison of the numerical results with analytic ones is made. The method is found to be stable using stability analysis and convergence criteria. The effect of Peclet number on exit solute concentration and other parameters is presented in the form of breakthrough curves. The results are derived for a broad range of parameters and the present method is found to be more useful and refined for solving the two-point boundary value problems.

This paper studies the interaction of an electromagnetic field with the matter in a laser cavity without assuming a fixed direction of the transverse electric field, described by the two-level Maxwell–Bloch equations. The derivation of the laser (3+1)-dimensional vectorial cubic-quintic complex Ginzburg–Landau equation is reported using a perturbative nonlinear analysis performed near the laser threshold. Considering the vector (2+1)D cubic-quintic complex Ginzburg–Landau equation, the stability of the moving dissipative solitons in the laser cavity is analyzed. Using the variational approximation, stability conditions and propagation trajectories of dissipative solitons are derived. Direct numerical simulations fully confirm analytical predictions of dissipative solitons trapped in an effective potential well. Potential applications of the obtained results related to spatial dissipative solitons, may be found in class B laser by considering solitons as individual addressable and shift registers of the all-optical data processing systems.

We investigate the modulational instability (MI) of a continuous wave (cw) under the combined effects of higher-order dispersions, self steepening and self-frequency shift, cubic, quintic, and septic nonlinearities. Using Maxwell's theory, an extended nonlinear Schr\"odinger equation is derived. The linear stability analysis of the cw solution is employed to extract an expression for the MI gain, and we point out its sensitivity to both higher-order dispersions and nonlinear terms. In particular, we insist on the balance between the sixth-order dispersion and nonlinearity, septic self-steepening, and the septic self-frequency shift terms. Additionally, the linear stability analysis of cw is confronted with the stability conditions for solitons. Different combinations of the dispersion parameters are proposed that support the stability of solitons and the occurrence of MI. This is confronted with full numerical simulations where the input cw gives rise to a broad range of behaviors, mainly related to nonlinear patterns formation. Interestingly, under the activation of MI, a suitable balance between the sixth-order dispersion and the septic self-frequency shift term is found to highly influence the propagation direction of the optical wave patterns.

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.

This paper deals with the trajectory planning for serial robotic manipulators passing through key points by minimizing execution time, energy consumption and joint jerks. Quintic NURBS curves are adopted to fit the trajectory, of which the trajectory is reparameterized with respect to time for generation of geometric path and motion laws, aiming at continuity of the robot velocity, acceleration and jerk. A trajectory planning approach for optimum robot performance is proposed by solving a multi-objective optimization problem to attain optimal curve parameters and distributed execution time along curve segments simultaneously. The proposed technique of trajectory planning is numerically illustrated with a robotic arm and evaluated by experimental measurements. The comparison of total execution time and joint dynamics with/without variables optimization shows the effectiveness of the proposed approach.

As for industrial robots’ point-to-point joint motion planning with constrained velocity, cubic polynomial planning has the problem of discontinuous acceleration, quintic polynomial planning requires acceleration to be specified in advance, which will likely cause velocity to fluctuate largely because appropriate acceleration assigned in advance is hardly acquired. Aiming at these problems, a modified cubic Hermite interpolation for joint motion planning was proposed. In the proposed methodology, knots of cubic Hermite interpolation need to be reconfigured according to the initial knots. The formulas for how to build new knots were put forward after derivation. Using the newly-built knots instead of initial knots for cubic Hermite interpolation, joint motion planning was carried out. The purpose was that the joint planning not only satisfied the displacement and velocity constraints at the initial knots but also guaranteed C2 continuity and less velocity fluctuation. A study case was given to verify the rationality and effectiveness of the methodology. Compared with the other two planning methods, it proved that the raised problems can be solved effectively via the proposed methodology, which is beneficial to the working performance and service life of industrial robots.

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 paper, a numerical scheme based on quintic B-spline collocation method using the Strang splitting technique is presented for solving the generalised Rosenau–regularised long wave (RLW) equation given by appropriate initial-boundary values. For this purpose, firstly the problem is split into two subproblems such that each one includes the derivative in the direction of time. Secondly, each subproblem is reduced to a system of ordinary differential equations (ODEs) using collocation finite-element method with quintic B-splines for spatial integration. Then, the resulting ODEs for time integration are solved using the Strang time-splitting technique with the second order via the usual Runge–Kutta (RK-4) algorithm with the fourth order. To measure the accuracy and efficiency of the present scheme, a model problem with an exact solution is taken into consideration and investigated for various values of the parameter p. The error norms $$L_{2}$$ and $$L_{\infty }$$ together with the invariants of discrete mass Q and discrete energy E have been computed and a comparison is given with other ones found in the literature. The convergence order of the present numerical scheme has also been computed. Furthermore, the stability analysis of the scheme is numerically examined.

We present a numerical investigation of the propagation dynamics of a truncated Airy pulse in a highly dispersive and nonlinear medium by employing the split-step Fourier transform method and look, in particular, into the effects of fourth order dispersion (FOD) and cubic-quintic-septic nonlinearity on pulse evolution. Presence of FOD cancels the Airy pulse self-acceleration along with eclipsing the oscillatory tail during propagation in the linear regime. Further, we observe soliton shedding at low input pulse power in the presence of cubic and quintic nonlinearity and negative FOD. The emergent soliton exhibits temporal shift and the direction and the extent of the shift depend upon the strengths of cubic and quintic nonlinearities. In the presence of anomalous group velocity dispersion (GVD) with negative FOD, soliton shedding is observed at relatively high input pulse power. The strengths of GVD and nonlinearity play a vital role in the temporal shifting of emergent soliton. Furthermore, we have explored the effects of septic nonlinearity on soliton shedding in different scenarios of nonlinearity and dispersion., 11 pages, 12 figures

We study an intense-short pulse propagation in a saturable cubic-quintic nonlinear media in the presence of nonlinear dispersion within the framework of an extended variational approach. We derive an effective equation for the pulse width and demonstrate how the saturation due to nonlinearity is achieved in presence of nonlinear dispersion. We find that the nonlinear dispersion can change the pulse width and induce motion in the system. The direction of induced motion depends on the sign of nonlinear dispersion. The pulse is energetically stable at an equilibrium width. A disturbance can, however, induce oscillation in pulse width, the frequency of which is always smaller due to nonlinear dispersion. We check dynamical stability by a direct numerical simulation., 7 pages, 5 figures

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 our previous paper we have elaborated a certain signed count of real lines on real projective n-dimensional hypersurfaces of degree 2n-1. Contrary to the honest "cardinal" count, it is independent of the choice of a hypersurface, and by this reason provides a strong lower bound on the honest count. In this count the contribution of a line is its local input to the Euler number of a certain auxiliary vector bundle. The aim of this paper is to present other, in a sense more geometric, interpretations of this local input. One of them results from a generalization of Segre species of real lines on cubic surfaces and another from a generalization of Welschinger weights of real lines on quintic threefolds., Comment: 20 pages, typos are corrected (most essential, in Proposition 4.3.3)

We consider the focusing energy-critical quintic nonlinear wave equation in 3D Euclidean space. It is known that this equation admits a one-parameter family of radial stationary solutions, called solitons, which can be viewed as a curve in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$, for any $s> 1/2$. By randomizing radial initial data in $ \dot H^s_x({{\mathbb{R}}}^3) \times H^{s-1}_x({{\mathbb{R}}}^3)$ for $s> 5/6$, which also satisfy a certain weighted Sobolev condition, we produce with high probability a family of radial perturbations of the soliton that give rise to global forward-in-time solutions of the focusing nonlinear wave equation that scatter after subtracting a dynamically modulated soliton. Our proof relies on a new randomization procedure using distorted Fourier projections associated to the linearized operator around a fixed soliton. To our knowledge, this is the 1st long-time random data existence result for a focusing wave or dispersive equation on Euclidean space outside the small data regime.

The aim of this paper is to show the small data scattering for 2D ICQNLS: i u t = − Δ u + K 1 ( x ) | u | 2 u + K 2 ( x ) | u | 4 u . Under the assumption that | ∂ j K l | ≲ | x | b l − j for j = 0 , 1 , 2 , l = 1 , 2 and 0 ≤ b l ≤ l − 2 3 , we prove the small data scattering in an angularly regular Sobolev space H θ 1 , 1 . We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space H θ 1 , 1 ⊂ H 1 with angular regularity such that ‖ ∂ θ f ‖ H 1 ∞ , and also use the recently developed angularly averaged Strichartz estimates (Tao, 2000; Cho and Lee, 2013; Guo et al., 2018). In addition, we suggest a sufficient condition for non-existence of scattering.

In this work, we introduce a two-dimensional domain predator-prey model with strong Allee effect and investigate the Turing instability and the phenomena of the emergence of patterns. The occurrence of the Turing instability is ensured by the conditions that are procured by using the stability analysis of local equilibrium points. The amplitude equations (for supercritical case cubic Stuart–Landau equation and for subcritical quintic Stuart–Landau equation) are derived appropriate for each case by using the method of multiple time scale and show that the system supports patterns like squares, stripes, mixed-mode patterns, spots and hexagonal patterns. We obtain the asymptotic solutions to the model close to the onset instability based on the amplitude equations. Finally, numerically simulations tell how cross-diffusion plays an important role in the emergence of patterns.

In the present work, the authors describe two B-spline collocation methods for a class of nonlinear singular Lane-Emden type equations which describe several phenomena in theoretical physics and astrophysics. The first method is based on non-optimal quintic B-spline collocation approach, while the second method is based on optimal quintic B-spline collocation approach. We note that the former method yields fourth-order convergent approximation to the solution of the second-order boundary value problems. To obtain an optimal convergence of order six to the solution of the same problem, the later method is designed by perturbing the original problem. Convergence results for both methods are established via Green's function approach. Four nonlinear examples are provided to illustrate the suggested methods and to verify their theoretical rates of convergence. It is observed that the first method yields non-optimal fourth order accuracy and the second method exhibits optimal sixth order accuracy for the solution of the problem. Moreover, numerical results appear to be higher accurate when compared to other methods.

This paper aims to focus on the implementation of a new approach, quintic trigonometric B-spline basis functions in differential quadrature method to find numerical solution of generalized fourth order partial differential equations with nonlinearity involved. The obtained results using this approach are presented in comparison with available exact and numerical solutions obtained by other researchers. The obtained solutions are in agreement with the available exact solutions and even better than the solutions proposed by the other schemes in the literature. The applicability of the scheme are demonstrated by different eight test problems for the discussed equations that are simulated, for calculating errors like L2, $${{L}_{\infty }}$$, RMS and GRE. To visualize the same, solutions are also presented graphically along with the exact solutions. Scheme is shown to be unconditionally stable with the help of eigenvalues, which demonstrates the consistency of the proposed numerical scheme.

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.

Studies on the water waves contribute to the design of the related industries, such as the marine and offshore engineering, while the media with the negative refractive index can be applied as the carrier media in fiber optics. In consideration of the inhomogeneities of the media and nonuniformities of the boundaries in the real physical backgrounds, a quintic time-dependent-coefficient derivative nonlinear Schrodinger equation for certain hydrodynamic wave packets or medium with the negative refractive index is investigated in this paper. Bilinear forms and the N-soliton solutions with respect to the nonzero background, which are different from those in the existing studies, are derived under the certain constraints. Conditions for the dark/anti-dark/gray solitons are deduced due to the properties of the solitons derived via the asymptotic analysis. Effects of the dispersion coefficient $$\lambda (t)$$ , self-steepening coefficient $$\alpha (t)$$ , cubic nonlinearity $$\mu (t)$$ and quintic nonlinearity $$\nu (t)$$ on the interactions between the anti-dark and gray solitons under the certain condition are investigated. Interactions among the dark, anti-dark and gray solitons are discussed under two cases: when $${\alpha (t)}/{\lambda (t)}$$ and $${\mu (t)}/{\lambda (t)}$$ are the constants, whether the interaction is elastic or not depends on whether $$\lambda (t)$$ , $$\alpha (t)$$ and $$\mu (t)$$ are the constants or the functions of t; when $${\alpha (t)}/{\lambda (t)}$$ and $${\mu (t)}/{\lambda (t)}$$ are related to t, if the velocity of the soliton is a periodic function of t, the propagation of the corresponding soliton is periodic and the corresponding interaction is inelastic. Interactions among the three/four solitons are described to be elastic or inelastic based on the changes in the velocities and waveforms of the three/four solitons after the interactions.

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.

Let f ( x ) = x 5 + a x 3 + b x 2 + c x , where a, b, and c are elements of the finite field k of size q = p t , where p > 5 and q ≡ 1 or 4 (mod 5). We determine those f ( x ) 's for which the Galois closure of k ( x ) / k ( f ( x ) ) is a rational function field with Galois group A 5 . We then determine the size of the value sets of these polynomials.

The usual way of solving a solvable quintic equation has been to establish more equations than unknowns, so that some relation among the coefficients comes up, leading to the solutions. In this paper, a relation among the coefficients of a principal quintic equation is established by effecting a change of variable and inserting a root to the quintic equation, and then equating odd-powers of the resulting sextic equation to zero. This leads to an even-powered sextic equation, or equivalently a cubic equation; thus one needs to solve the cubic equation. We break from this tradition, rather factor the even-powered sextic equation in a novel fashion, such that the inserted root is identified quickly along with one root of the quintic equation in a quadratic factor of the form, u 2 − g 2 = (u + g)(u − g). Thus there is no need to solve any cubic equation. As an extra benefit, this root is a function of only one coefficient of the given quintic equation.

Under parity-time symmetric potentials, different-order nonlinearities such as cubic, quintic and septimal nonlinearities, altogether with their combinations and second-order and fourth-order dispersions/diffractions are simultaneously considered to form three-dimensional optical solitons. Based on some high-order nonlinear Schrodinger equations, three-dimensional analytical optical soliton solutions are found. In the defocusing cubic nonlinear case, three-dimensional optical soliton without fourth-order diffraction/dispersion is stable than that with fourth-order diffraction/dispersion. However, in the defocusing cubic and focusing quintic nonlinear case, the stability situation of soliton is just on the contrary. Among all combinations of nonlinearity, the stability of three-dimensional optical soliton in the cubic-quintic nonlinear case is better than that in the cubic nonlinear case, but worse than that in the cubic-quintic-septimal nonlinear case. In the quintic-septimal nonlinear case, three-dimensional optical soliton is unstable and will collapse ultimately.

The problem of determining necessary and sufficient conditions on P and Q for system. x = - y+P(x+y),y = x+Q(x+y) to have a center at the origin is known as the Poincaré center-focus problem. So far, people has tried many ways to solve the problem of central focus. However, it is difficult to solve the center focus problem of higher order polynomial system. In this paper, we use the Poincaré and Alwash-Lloyd methods to study the center focus problem and derive the center conditions of the five periodic differential equation.

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.

In this paper, we obtain optical soliton solutions for non-Kerr law nonlinear Schrodinger equation (NLSE) with third order (3OD) and fourth order dispersions (4OD). We will use two integration schemes, namely sin-cosine method and Bernoulli’s equation approach with five laws of nonlinearities. Sine-cosine method is applicable to Kerr, power and anti-cubic laws, this method provides bright soliton solutions. The second method is applicable to parabolic and cubic quintic laws, this method generates dark soliton. The results may be used in discussing the propagation of optical solitons in highly dispersive media with Kerr, power, anti-cubic, parabolic and cubic quintic law nonlinearities.

The structure and the production process for flying shear machine are introduced first. Then, a quintic polynomial is applied to the design of an electronic cam system for the rotary knife axis in short materials cutting. The dimensionless equation for a quintic polynomial cam curve is deduced. Finally, the curve is plotted with the cam constructor integrated into Siemens engineering development software SCOUT and it is tested with a laboratory platform, which consists of a motion controller SIMOTION and motor drivers SINAMICS S120. The results show that the running stability of the flying shear machine and the position control accuracy of the rotary knife can be effectively improved by using the curve designed in this paper.

In this paper, a dynamic automatic obstacle avoidance trajectory planning and tracking control framework is proposed for tractor-trailer system. Tractor-trailer is a special class of multibody and nonholonomic system, whose backward and forward operations have difference kinetic mechanisms. Because the obstacle avoidance behaviors are concerned with the two motion modes, the kinematic models including backward and forward movements are firstly derived. Secondly, a time-based quintic polynomial function is developed to plan two kinds of dynamic obstacle avoidance trajectories based on dynamics constraints and the information from on board sensors, so as to minimize the collision risk. Thirdly, a model predictive control (MPC)-based posture controller is designed, by which better tracking performance can be achieved for both forward and backward obstacle avoidance maneuvers. Lastly, the simulation results validate the effectiveness of the proposed dynamic obstacle avoidance framework and the designed methods.

We prove that a curve of degree dk on a very general surface of degree d ≥ 5 in P 3 has geometric genus at least d k ( d − 5 ) + k 2 + 1 . This gives a substantial improvement on the celebrated genus bounds of Geng Xu. As a corollary, we deduce the algebraic hyperbolicity of a very general quintic surface in P 3 , resolving a long-standing conjecture of Demailly. This completely determines which very general hypersurfaces in P 3 are algebraically hyperbolic.

In this study, fractional differential equations having quintic nonlinearity are considered by proposing an accurate numerical method based on the matching polynomial and matrix-collocation system. This method provides an integration between matrix and fractional derivative, which makes it fast and efficient. A hybrid computer program is designed by making use of the fast algorithmic structure of the method. An error analysis technique consisting of the fractional-based residual function is constructed to scrutinize the precision of the method. Some error tests are also performed. Figures and tables present the consistency of the approximate solutions of highly stiff model problems. All results point out that the method is effective, simple, and eligible. © 2019 John Wiley & Sons, Ltd.

We study the Clifford type inequality for a particular type of curves $C_{2,2,5}$, which are contained in smooth quintic threefolds. This allows us to prove some stronger Bogomolov-Gieseker type inequalities for Chern characters of stable sheaves and tilt-stable objects on smooth quintic threefolds. Employing the previous framework by Bayer, Bertram, Macr\`i, Stellari and Toda, we construct an open subset of stability conditions on every smooth quintic threefold in $\mathbf{P}^4_{\mathbb C}$., Comment: pre-journal version, 32 pages, 7 figures, comments are very welcome!

In this paper, the nonlinear Bratu type equation is numerically solved via quintic B-spline method. It is shown that the proposed technique has fourth order convergence. The efficiency and ...

In this paper, a nonlinear active vibration control method, which is suitable for the vibration mitigation of both linear and nonlinear structures, is proposed. The main idea of this method is to make the controller emulate a pendulum-type auto-parametric vibration absorber, with the purpose of counteracting the different types of nonlinearity in structures. In this way, the difficulty in the passive realization of the pendulum-type vibration absorber due to the physical unsuitable parameters can also be well released. The idea of introducing an exosystem in Byrnes–Isidori regulator is borrowed to design the nonlinear controller. The steps to choose the parameters of the controller are introduced. The validity and robustness of the controller for the vibration mitigation of both linear and nonlinear oscillators are investigated by numerical examples of application. The simulation results show that the proposed nonlinear controller can effectively mitigate the vibration not only of linear oscillators, but also of nonlinear oscillators with third-, quintic- and seventh-order nonlinearities. The good robustness of the proposed nonlinear controller is also demonstrated.

Various physical, ecological, economic, etc phenomena are governed by planar differential systems. Sub- sequently, several research studies are interested in the study of limit cycles because of their interest in the understanding of these systems. The aim of this paper is to investigate a class of quintic Kolmogorov systems, namely systems of the form x=xP4 (x;y); y= y Q4 (x; y) ; where P4 and Q4 are quartic polynomials. Within this class, our attention is restricted to study the limit cycle in the realistic quadrant {(x; y) 2 R2; x > 0; y > 0}. According to the hypothesises, the existence of algebraic or non-algebraic limit cycle is proved. Furthermore, this limit cycle is explicitly given in polar coordinates. Some examples are presented in order to illustrate the applicability of our result

It is usually believed that solitons always expand due to absorption, and vice versa, they can be significantly compressed due to amplification. Contrary to this usual opinion, we demonstrate that the higher-order nonlinearity opens the possibility to observe unusual and contrasting effects, namely, the soliton self-spreading effect arising due to the soliton amplification and vise versa, the soliton self-compression effect appearing in the absorption regime of the soliton propagation. The detailed contrasting scenarios of the soliton self-spreading or self-compression owing to amplification or absorption are considered in the framework of the simplest cubic-quintic nonlinear Schrodinger equation model.