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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.

We study renormalization group multicritical fixed points in the ϵ-expansion of scalar field theories characterized by the symmetry of the (hyper)cubic point group HN. After reviewing the algebra of HN-invariant polynomials and arguing that there can be an entire family of multicritical (hyper)cubic solutions with ϕ2n interactions in $$ d=\frac{2n}{n-1}-\epsilon $$ d = 2 n n − 1 − ϵ dimensions, we use the general multicomponent beta functionals formalism to study the special cases d = 3 − ϵ and $$ d=\frac{8}{3}-\epsilon $$ d = 8 3 − ϵ , deriving explicitly the beta functions describing the flow of three- and four-critical (hyper)cubic models. We perform a study of their fixed points, critical exponents and quadratic deformations for various values of N, including the limit N = 0, that was reported in another paper in relation to the randomly diluted single-spin models, and an analysis of the large N limit, which turns out to be particularly interesting since it depends on the specific multicriticality. We see that, in general, the continuation in N of the random solutions is different from the continuation coming from large-N, and only the latter interpolates with the physically interesting cases of low-N such as N = 3. Finally, we also include an analysis of a theory with quintic interactions in $$ d=\frac{10}{3}-\epsilon $$ d = 10 3 − ϵ and, for completeness, the NNLO computations in d = 4 − ϵ.

We classify the centers of the polynomial differential systems in R2 of degree d ≥ 5 odd that in complex notation writes as z˙ = iz + (zz¯)d−5/2 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5), where A, B, C, D, E, F ∈ C and either A = Re(D) = 0, or A = Im(D) = 0, or Re(A) = D = 0, or Im(A) = D = 0. The first author is partially supported by a MINECO/ FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2014SGR 568, ICREA Academia, two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378. The third author is supported by Portuguese National Funds through FCT – Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD.

In this paper, we establish the Hyers-Ulam stability of quintic functional equation in F-spaces by using direct method.

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.

Different from urban environment with high-precision map, off-road environment lacks rich priori road structure information and precise location information, which is a huge challenge for autonomous driving. To solve these problems, this paper proposes a real-time decision-making and local trajectory planning method based on a predefined global reference line. According to the reference line, a cluster of parallel offset curves with corresponding obstacle and deviation costs are generated. Then, an appropriate parallel line with minimum cost is selected as the preliminary decision, and the target position with a matched speed on this line is choosed for local planning. Finally, the lateral and longitudinal trajectories planning is carried out with the initial constraint and target constraint of the quintic polynomials. The proposed method is verified in a 3.6 km bumpy off-road environment, including straight road, curves, continuous U-turn and other driving conditions. The maximum autonomous driving speed of an small-medium-sized platform can reach 30 km/h under the condition of stable and smooth driving.

In the cam design process, cam motion functions play an important role due to affect not only kinematics but also the dynamics of the cam-follower systems. However, the selection of cam motion functions has been poor so far. This study aims to evaluate B-spline for the synthesis of cam motion with a large number of output conditions. The procedure for determining the cam motion was investigated. In addition, the comparison of the motion diagram between the cubic and quintic B-spline was also proposed. Moreover, Fourier analysis was used to examine the response of the kinematics in cam-follower systems. The results demonstrate that using quintic B-spline can be given good characteristics of the cam motion.

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.

This paper presents a real-time joint trajectory interpolation system for the purpose of frequency scaling the low cycle time of a robot controller, allowing a Python application to real-time control the robot at a moderate cycle time. Interpolation is based on quintic Hermite piece-wise splines. The splines are calculated in real-time, in a piecewise manner between the high-level, long cycle time trajectory points, while sampling of these splines at an appropriate, shorter cycle time for the real-time requirement of the lower-level system. The principle is usable in general, and the specific implementation presented is for control of the Panda robot from Franka Emika. Tracking delay analysis is presented based on a cosine trajectory. A simple test application has been implemented, demonstrating real-time feeding of a pre-calculated trajectory for cutting with a knife. Estimated forces on the robot wrist are recorded during cutting and presented in the paper.

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.

In this paper, we shall solve a time-fractional nonlinear Schrödinger equation by using the quintic non-polynomial spline and the L1 formula. The unconditional stability, unique solvability and convergence of our numerical scheme are proved by the Fourier method. It is shown that our method is sixth order accurate in the spatial dimension and $(2-\gamma )$ ( 2 − γ ) th order accurate in the temporal dimension, where γ is the fractional order. The efficiency of the proposed numerical scheme is further illustrated by numerical experiments, meanwhile the simulation results indicate better performance over previous work in the literature.

We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all $2$-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two ordinary double points and provide explicit expressions that allow one to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using $A$-discriminants and intersection theory. Finally, we show that toric Fano varieties associated to 3-valent phylogenetic trees have ML degree one and provide a formula for the maximum likelihood estimate. We prove it as a corollary to a more general result about the multiplicativity of ML degrees of codimension zero toric fiber products, and it also follows from a connection to a recent result about staged trees., Comment: 28 pages, 4 figures, 4 tables, this article supersedes arXiv:1602.08307

Applying the Slime Mould Algorithm (SMA) in optimization and coding it with Python programming language, we found that runtime warnings occurred because of the inherent limits of arctanh functions. Considering the circumstances that the maximum residual errors would be constrained and the maximum allowed iteration number would be bothering, we therefore proposed an improvement to the SMA with exponential functions. Three kinds of benchmark functions were introduced to verify the suitability, accessibility, and capability. Experiment results demonstrated that the improved SMA with exponential functions would be suitable and accessible. Comparing with the standard SMA, the improved algorithm would result in better performance with exception that they perform similar when optimizing the Quintic function, the representative of those kinds of benchmark functions with basins in their three-dimensional profiles.

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.

Considering the theory of electromagnetic waves from the Maxwell's equations, we introduce a (3+1)-dimensionsal cubic-quintic complex Ginzburg-Landau equation describing the dynamics of dissipative light bullets in nonlinear metamaterials. The model equation, which is derived beyond the slowly varying envelope approximation, includes the effects of diffraction, dispersion, loss, gain, cubic, and quintic nonlinearities, as well as cubic and quintic self-steepening effects. The modulational instability of the plane waves is studied both theoretically, using the linear stability analysis, and numerically, using direct simulations of the Fourier space of the proposed nonlinear wave equation, based on the Drude model. The linear theory predicts instability for any amplitude of the primary wave. Also, in the linear stability analysis, self-steepening effects of different orders are confronted and one discusses their effects on the behavior of the gain spectrum under both normal and anomalous group-velocity dispersion regimes. Analytical results are equally confronted to direct numerical simulations and fully agree with the predictions from the gain spectra. Modulational instability is manifested by clusters of solitons and multihump and dromion-like structures, whose emergence and features depend not only on system parameters, such as the cubic and quintic self-steepening coefficients, but also on the propagation distance under a suitable balance between nonlinear and dispersive effects.

Many soft robotic systems for manipulation have been developed for their compliance and their smooth obstacles avoidance capabilities contrary to rigid systems. In order to increase human task aid, a new class of robots known as mobile soft manipulators has thus emerged, where the path planning of the whole system, both soft manipulator and its mobile platform, remains a real scientific challenge. For that purpose, in this paper, we suggest a curvature control algorithm for that class of hyper-redundant mobile soft manipulator approaching and handling objects with simultaneous obstacle avoidance based on potential field of velocity. The algorithm is based on principle of sliding mode, which converges in a finite time. The soft manipulator kinematics is modeled in 2D by the mean of Pythagorean Hodograph (PH) quintic curve theory with length constraints. The result of this modeling and control is applied on Festo Robotino-XT, mobile and soft manipulator robot.

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.

In this paper, we describe a general spline maximum entropy method for the approximation of solutions for solving Fredholm integral equations. The general spline maximum entropy method allows to extract piecewise linear, piecewise quadratic, piecewise cubic, piecewise quartic, piecewise quintic and other maximum entropy methods for solving Fredholm integral equations. We present numerical examples of piecewise linear, piecewise quadratic, piecewise cubic, piecewise quartic, piecewise quintic and other piecewise maximum entropy methods for solving Fredholm integral equations of second kind and first kind respectively. A proof of convergence for the piecewise spline maximum entropy method is also presented.

We argue that the conjectural relation between the subleading term in the small-squashing expansion of the free energy of general three-dimensional CFTs on squashed spheres and the stress-tensor three-point charge $t_4$ proposed in arXiv:1808.02052: $F_{\mathbb{S}^3_{\varepsilon}}^{(3)}(0)=\frac{1}{630}\pi^4 C_{\scriptscriptstyle T} t_4$, holds for an infinite family of holographic higher-curvature theories. Using holographic calculations for quartic and quintic Generalized Quasi-topological gravities and general-order Quasi-topological gravities, we identify an analogous analytic relation between such term and the charges $t_2$ and $t_4$ valid for five-dimensional theories: $F_{\mathbb{S}^5_{\varepsilon}}^{(3)}(0)=\frac{2}{15}\pi^6 C_{ \scriptscriptstyle T} \left[1+\frac{3}{40} t_2+\frac{23}{630} t_4\right]$. We test both conjectures using new analytic and numerical results for conformally-coupled scalars and free fermions, finding perfect agreement., Comment: 52 pages, 2 figures

Many fields of science rely on the collection of samples and estimation of true population distributions from those samples. There are several effective nonparametric methods for approximating a true distribution from empirical data, however it is unclear which methods produce the best approximations in practice. This work presents a case study on the effectiveness of various distribution approximations. Results show that piecewise linear approximations produce the smallest maximum absolute error, while the classic empirical distribution function (EDF) produces the smallest median absolute error as well as the smallest first quartile and minimum absolute error when approximating a distribution from a sample. When building distribution prediction models, the piecewise quintic and cubic approximations produce the lowest absolute error at most error percentiles. This case study encourages more research on the best methods of fitting empirical data with smooth functions to generate accurate distribution approximations.

This paper deals with Pythagorean hodograph curves of the spaces of algebraic trigonometric functions and trigonometric polynomials, $\text {span}\left \{1,t,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \} $ and $ \text {span}\left \{1,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \}, $ respectively. First, we propose a general characterization of planar PH curves in these spaces. Next, we consider the particular cases m = 1 and m = 2. For each of them, we give the general form of the control polygon of the PH curves, the implicit relations defining these curves, and their geometrical interpretations. Some examples and particular cases complete this study.

This article aims to present a minimum-jerk trajectory planning approach to address the smooth trajectory generation problem of 3-prismatic-universal-universal translational parallel kinematic manipulator. First, comprehensive kinematics and dynamics characteristics of this 3-prismatic-universal-universal parallel kinematic manipulator are analyzed by virtue of the accepted link Jacobian matrices and proverbial virtual work principle. To satisfy indispensable continuity and smoothness requirements, the discretized piecewise quintic polynomials are employed to interpolate the sequence of joints’ angular position knots which are transformed from these predefined via-points in Cartesian space. Furthermore, the trajectory planning problem is directly converted into a constrained nonlinear multi-variables optimization problem of which objective function is to minimize the maximum of the joints’ angular jerk throughout the whole trajectory. Finally, two typical application simulations using the reliable sequential quadratic programming algorithm demonstrate that this proposed minimum-jerk trajectory planning approach is of explicit feasibility and appreciable effectiveness.

We study the dynamics of sonic black hole horizon formation of quasi-one-dimensional (1D) Bose-Einstein condensate incorporating higher-order quintic nonlinear interaction. Based on the one dimensional Gross-Pitaevskii equation with nonlinearity up to the quintic order, we derived a novel analytical formula for the key dynamical variables of sonic horizon formation using the modified variational method and exact F-expansion method. We obtained good agreement between the key dynamical variables from the two different methods. The stabilization effects of higher-order nonlinear interaction along with the more precise location of the sonic horizon boundary were illustrated. The theoretical results obtained in this work can be used to guide relevant experimental observations of sonic black hole-related dynamics incorporating the effects of higher-order nonlinear interaction.

The numerical solutions of the coupled Korteweg-de Vries equation is investigated via combination of two efficient methods that includes Crank–Nicolson scheme for time integration and quintic B-spline based differential quadrature method for space integration. The differential quadrature method has the advantage of using less number of grid points. And the advantage of the Crank–Nicolson scheme is the prevention of long and tedious algebraic computations for time integration. Those advantages come together and produce better results. To display the accuracy and efficiency of the present hybrid method three well-known test problems, namely single soliton, interaction of two soliton and birth of solitons are solved and the error norms $$L_{2}$$ and $$ L _{\infty }$$ are computed and compared with earlier works. Present hybrid method obtained superior results than earlier works by using the same parameters and less number of grid points. This situation is shown by comparison of the earlier works. At the same time, two lowest invariants and numerical and analytical values of the amplitudes of the solitons during the simulations are computed and tabulated. Besides those, relative changes of invariants are computed. Properties of solitons observed clearly at the all of the test problems and figures of the all of the simulations are given.

This paper presents a new trajectory planning method based on the improved quintic B-splines curves for a three degrees-of-freedom (3-DOF) cable-driven parallel robot (CDPR). First, the conditions of positive cables’ tension are expressed in terms of the position and acceleration constraints of the end-effector. Then, an improved B-spline curve is introduced, which is employed for generating a pick-and-place path by interpolating a set of given via-points. Meanwhile, by expressing the position and acceleration of the end-effector in terms of the first and second derivatives of the improved B-spline, the cable tension constraints are described in the form of B-spline parameters. According to the properties of the defined pick-and-place path, the proposed motion profile is dominated by two factors: the time taken for the end-effector to pass through all the via-points and the ratio between the nodes of B-spline. The two factors are determined through multi-objective optimization based on the efficiency coefficient method. Finally, experimental results on a 3-DOF CDPR show that the improved B-spline exhibits overall superior behavior in terms of velocity, acceleration, and cables force compared with the traditional B-spline. The validity of the proposed trajectory planning method is proved through the experiments.

In this paper, bright-dark, multi solitons, and other solutions of a (3 + 1)-dimensional cubic-quintic complex Ginzburg&ndash, Landau (CQCGL) dynamical equation are constructed via employing three proposed mathematical techniques. The propagation of ultrashort optical solitons in optical fiber is modeled by this equation. The complex Ginzburg&ndash, Landau equation with broken phase symmetry has strict positive space&ndash, time entropy for an open set of parameter values. The exact wave results in the forms of dark-bright solitons, breather-type solitons, multi solitons interaction, kink and anti-kink waves, solitary waves, periodic and trigonometric function solutions are achieved. These exact solutions have key applications in engineering and applied physics. The wave solutions that are constructed from existing techniques and novel structures of solitons can be obtained by giving the special values to parameters involved in these methods. The stability of this model is examined by employing the modulation instability analysis which confirms that the model is stable. The movements of some results are depicted graphically, which are constructive to researchers for understanding the complex phenomena of this model.

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.

Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$ embeds symplectically in a quintic surface, then $p \leq 12$, partially answering the symplectic version of a question of Kronheimer., 34 pages. v2: minor referee's corrections and bibliographic updates. This version to appear in Algebraic Geometry

Recently, many researchers established various methods to construct optical solutions in the field of nonlinear optics because of optical solitons which shape the fundamental component to transport data from side to side of the earth for very wide distances. There are countless styles that effectively define this movement of soliton construction.1–30 In this paper, we investigate the derivative nonlinear Schr¨odinger’s equation (DNLSe) with the dimensionless shape9,10 to construct optical solitons including traveling wave solution (TWS) using the extended generalizing Riccati mapping (EGRM). The extended tanh function method is further improved by the EGRM and the constructed optical solitons. The attained solitons are represented in four families containing trigonometric functions, rational functions and hyperbolic functions. The method is presented together with the (G0/G)-expansion method and this process is an influential mathematical implement for solving NPDEs. Essentially, on in this method, the second-order linear ODE with constant coefficients is considered as an auxiliary equation. The auxiliary equation G0 (φ) = h + fG(φ) + gG2 (φ) is used, where f, g and h are arbitrary constants. The familiar Kaup–Newell’s equation refers to the mechanical functions of subpico second optical soliton movement.9,10 There are many studies for this model in several fields. Also, this model is named as first of the three forms of derivative NLSE. In Ref. 10, Triki and Biswas generalized the Kaup–Newell’s equation (KNe). They studied their conserved quantities and obtained sub-pico second bright, singular and dark optical solitons. The DNLSe with the dimensionless shape is presented by as9 iqt + aqxx + ib(|q| 2n q)x = 0 . (1.1) The function q(x, t) is a complex-valued-dependent variable that defines wave profile. In Eq. (1.1), the first notation is the process notation, the second notation gives dispersal group speed and the third notation refers to the dispersal nonKerr notation, for n > 2. For n = 1, Eq. (1.1) represents the KNe. For n = 2, the differential quintic non-Kerr nonlinearity notations express importance in extension of very short impulses of width around sub-10 fs in extremely nonlinear optical fibers.

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.

We study a nonintegrable modified Korteweg-de-Vries equation containing a combination of third- and fifth-degree nonlinear terms that simulate waves in a three-layer fluid, as well as in spatially one-dimensional nonlinear-elastic deformable systems. It is established that this equation passes the Painleve test in a weak form. After the traveling wave transformation, this equation reduces to a generalized Weierstrass elliptic function equation, the right side of which is determined by a sixth-order polynomial in the dependent variable. Determined by the structure of the polynomial roots, the general solution of the equation is expressed in terms of the Weierstrass elliptic function or its successive degenerations—rational functions depending on the exponential functions of the traveling wave variable or directly on traveling wave variable. The classification of exact solitary-wave and periodic solutions is carried out, and the ranges of parameters necessary for their physical feasibility are revealed. An approach is proposed for constructing approximate solitary-wave and periodic solutions to generalized Weierstrass elliptic equation with a polynomial right-hand side of high orders.

A novel algorithm for computing monotone order six piecewise polynomial interpolants is proposed. Algebraic constraints for enforcing monotonicity are provided that align with quintic monotonicity theory. The algorithm is implemented, tested, and applied to several sample problems to demonstrate the improved accuracy of monotone quintic spline interpolants compared to the previous state-of-the-art monotone cubic spline interpolants.

In this note we study homological cycles in the mirror quintic Calabi-Yau threefold which can be realized by special Lagrangian submanifolds. We have used Picard-Lefschetz theory to establish the monodromy action and to study the orbit of Lagrangian vanishing cycles. For many prime numbers $p$ we can compute the orbit modulo $p$. We conjecture that the orbit in homology with coefficients in $\mathbb{Z}$ can be determined by these orbits with coefficients in $\mathbb{Z}_p$., Comment: 14 pages

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain (2005) on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract "counter-term theorem" `a la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find "many more" almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs., Comment: 44 pages

We use tropical curves and toric degeneration techniques to construct closed embedded Lagrangian rational homology spheres in a lot of Calabi-Yau threefolds. We apply this construction to the tropical curves obtained from the 2875 lines on the quintic Calabi-Yau threefold. Each admissible tropical curve gives a Lagrangian rational homology sphere in the corresponding mirror quintic threefold and disjoint curves give pairwise homologous but non-Hamiltonian isotopic Lagrangians. We check in an example that $>300$ mutually disjoint curves (and hence Lagrangians) arise. We show that the weight of each of these Lagrangians equals to the multiplicity of the corresponding tropical curve., 75 pages, 13 figures, final version, to appear in Forum Math. Sigma, ancillary files with source code and documentation added

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $\omega\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $\omega \in (-\infty,0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies $\omega_0$ and $\omega_1$ such that the standing waves are the ground states for $\omega \in [\omega_0,0)$, local minimizers of the energy at constant mass for $\omega \in (\omega_1,\omega_0)$, and saddle points of the energy at constant mass for $\omega \in (-\infty,\omega_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations., Comment: 30 pages; 3 figures

The problem of the optimal approximation of circular arcs by parametric polynomial curves is considered. The optimality relates to the Hausdorff distance and has not been studied yet in the literature. Parametric polynomial curves of low degree are used and a geometric continuity is prescribed at the boundary points of the circular arc. A general theory about the existence and the uniqueness of the optimal approximant is presented and a rigorous analysis is done for some special cases for which the degree of the polynomial curve and the order of the geometric smoothness differ by two. This includes practically interesting cases of parabolic G 0 , cubic G 1 , quartic G 2 and quintic G 3 interpolation. Several numerical examples are presented which confirm theoretical results.

We study the amplitude dynamics of two-dimensional (2D) solitons in a fast collision described by the coupled nonlinear Schr\"odinger equations with a saturable nonlinearity and weak nonlinear loss. We extend the perturbative technique for calculating the collision-induced dynamics of two one-dimensional (1D) solitons to derive the theoretical expression for the collision-induced amplitude dynamics in a fast collision of two 2D solitons. Our perturbative approach is based on two major steps. The first step is the standard adiabatic perturbation for the calculations on the energy balance of perturbed solitons and the second step, which is the crucial one, is for the analysis of the collision-induced change in the envelope of the perturbed 2D soliton. Furthermore, we also present the dependence of the collision-induced amplitude shift on the angle of the two 2D colliding-solitons. In addition, we show that the current perturbative technique can be simply applied to study the collision-induced amplitude shift in a fast collision of two perturbed 1D solitons. Our analytic calculations are confirmed by numerical simulations with the corresponding coupled nonlinear Schr\"odinger equations in the presence of the cubic loss and in the presence of the quintic loss., Comment: 25 pages, 6 figures

Research on meromorphic solution of complex differential equations using Nevanlinna theory has become a subject of great interest. This paper is devoted to finding meromorphic solutions of the cubic-quintic Ginzburg–Landau equation which arises in problems of dynamics, especially fluid dynamics. We consider the complex differential equation corresponding to cubic-quintic Ginzburg–Landau equation with coefficients being small functions of meromorphic functions. The problem has been mainly studied under the condition that meromorphic function and its first derivative share one value of the type counting multiplicity or ignoring multiplicity.

We present exact explicit Peregrine soliton solutions based on a periodic-wave background caused by the interference in the vector cubic-quintic nonlinear Schrödinger equation involving the self-steepening effect. It is shown that such periodic Peregrine soliton solutions can be expressed as a linear superposition of two fundamental Peregrine solitons of different continuous-wave backgrounds. Because of the self-steepening effect, some interesting Peregrine soliton dynamics such as ultrastrong amplitude enhancement and rogue wave coexistence are still present when they are built on a periodic background. We numerically confirm the stability of these analytical solutions against non-integrable perturbations, i.e., when the coefficient relation that enables the integrability of the vector model is slightly lifted. We also demonstrate the interaction of two Peregrine solitons on the same periodic background under some specific parameter conditions. We expect that these results may shed more light on our understanding of the realistic rogue wave behaviors occurring in either the fiber-optic telecommunication links or the crossing seas.

With the help of computer algebra system-\textsc{Mathematica}, this paper considers the weak center problem and local critical periods for bi-center of a $Z_2$-Equivariant quintic system with eight parameters. The order of weak bi-center is identified and the exact maximum number of bifurcation of critical periods generated from the bi-center is given via the combination of symbolic calculation and numerical analysis., Comment: 15 pages, 4 figures

We consider the focusing energy-critical inhomogeneous nonlinear Schrodinger equation: $$\displaystyle iu_t + \Delta u + g|u|{ }^4u = 0, \;\;u(0) \in \dot {H}^1,\;\; g \ge 0. $$ On the road map of Kenig–Merle [20] we show the global well-posedness and scattering of radial solutions under scaling, variational, and rigidity assumptions for g. We also provide sharp finite time blowup results for nonradial and radial solutions. For this we utilize the localized virial identity.

The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, and unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period as well as the exact analytic solution is given in terms of the famous Weierstrass elliptic function. An integrable case of a damped cubic-quintic equation is presented. Mathematica code for solving both cubic and cubic-quintic Duffing equations is given in Appendix at the end.

We prove a new smoothing type property for solutions of the 1d quintic Schr\"odinger equation. As a consequence, we prove that a family of natural gaussian measures are quasi-invariant under the flow of this equation. In the defocusing case, we prove global in time quasi-invariance while in the focusing case because of a blow-up obstruction we only get local in time quasi-invariance. Our results extend as well to generic odd power nonlinearities., Comment: Presentation improved

The construction of continuous adapted orthonormal frames along C 1 closed–loop spatial curves is addressed. Such frames are important in the design of periodic spatial rigid–body motions along smooth closed paths. The construction is illustrated through the simplest non–trivial context — namely, C 1 closed loops defined by a single Pythagorean–hodograph (PH) quintic space curve of a prescribed total arc length. It is shown that such curves comprise a two–parameter family, dependent on two angular variables, and they degenerate to planar curves when these parameters differ by an integer multiple of π. The desired frame is constructed through a rotation applied to the normal–plane vectors of the Euler–Rodrigues frame, so as to interpolate a given initial/final frame orientation. A general solution for periodic adapted frames of minimal twist on C 1 closed–loop PH curves is possible, although this incurs transcendental terms. However, the C 1 closed–loop PH quintics admit particularly simple rational periodic adapted frames.

We study the periodic cubic derivative nonlinear Schrodinger equation (DNLS) and the (focussing) quintic nonlinear Schrodinger equation (NLS). These are both $$L^2$$ critical dispersive models, which exhibit threshold-type behavior, when posed on the line $${{\mathbb {R}}}$$ . We describe the (three-parameter) family of non-vanishing bell-shaped solutions for the periodic problem, in closed form. The main objective of the paper is to study their stability with respect to co-periodic perturbations. We analyze these waves for stability in the framework of the cubic DNLS. We provide criteria for stability, depending on the sign of a scalar quantity. The proof relies on an instability index count, which in turn critically depends on a detailed spectral analysis of a self-adjoint matrix Hill operator. We exhibit a region in parameter space, which produces spectrally stable waves. We also provide an explicit description of the stability of all bell-shaped traveling waves for the quintic NLS, which turns out to be a two-parameter subfamily of the one exhibited for DNLS. We give a complete description of their stability—as it turns out some are spectrally stable, while other are spectrally unstable, with respect to co-periodic perturbations.

TWe suggest a method to construct exactly solvable Gross-Pitaevskii type equations, especially the variable-coefficient high-order Gross-Pitaevskii type equations. We show that there exists a relation between the Gross-Pitaevskii type equations. The Gross-Pitaevskii equations connected by the relation form a family. In the family one only needs to solve one equation and other equations in the family can be solved by a transform. That is, one can construct a series of exactly solvable Gross-Pitaevskii type equations from one exactly solvable Gross-Pitaevskii type equation. As examples, we consider the family of some special Gross-Pitaevskii type equations: the nonlinear Schr\"odinger equation, the quintic Gross-Pitaevskii equation, and cubic-quintic Gross-Pitaevskii equation. We also construct the family of a kind of generalized Gross-Pitaevskii type equation.

This paper discusses scattered data interpolation by using cubic Timmer triangular patches. In order to achieve C1 continuity everywhere, we impose a rational corrected scheme that results from convex combination between three local schemes. The final interpolant has the form quintic numerator and quadratic denominator. We test the scheme by considering the established dataset as well as visualizing the rainfall data and digital elevation in Malaysia. We compare the performance between the proposed scheme and some well-known schemes. Numerical and graphical results are presented by using Mathematica and MATLAB. From all numerical results, the proposed scheme is better in terms of smaller root mean square error (RMSE) and higher coefficient of determination (R2). The higher R2 value indicates that the proposed scheme can reconstruct the surface with excellent fit that is in line with the standard set by Renka and Brown&rsquo, s validation.