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We present an alternative relatively easy way to understand and determine the zeros of a quintic whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous \textit{Vorlesungen uber das Ikosaeder und die Auflosung der Gleichungen vom funften Grade} are here shortened via Heymann's theory of transformations. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series. Within this framework, we develop a practical algorithm to compute the zeros of the quintic.

The nonlinear Schrodinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrodinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results may be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as optical fibers, oceanography and so on.

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.

According to Galois’ theory, a solution x for the quintic equation cannot be solved for in radicals. Here we do not solve for x. We construct it. We note that radical numbers a and b can be found, such that a line that connects the points (a; f(a)) and (b; f(b)) passes through the interior point (x; 0), a root of the quintic function f(x). We then test this on f(x) = x5 +x2 + x+1/π. The values found for a and b, converted to decimal form, are a = −0.6012227544458956 and b = −0.6012348965947528, from which it is determined that x = −0.6012235335340363. The corresponding value, obtained through the software Mathematica numerical routines, is xn = −0.6012235335340362

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.

We derive the expressions for the collision-induced amplitude dynamics of two flat-top solitons in spatial dimensions of 1, 2, and 3 caused by the generic weak nonlinear loss under the framework of coupled cubic-quintic ( n + 1 )D nonlinear Schrodinger equations for n = 1 , 2 , 3 . We develop a new perturbative technique which is mainly based on the calculations for the fast collision-induced changes in the soliton envelope and the use of the single perturbed soliton solution for the calculations. These results quantify the energy dropdown due to a fast collision of two pulses ( n = 1 ) , or two optical beams ( n = 2 ), or two light bullets ( n = 3 ) in cubic-quintic nonlinear media with dissipation. The theoretical calculations are then confirmed by numerical simulations with the corresponding coupled nonlinear Schrodinger equations. Our approach can be used for studying the effects of dissipation on colliding solitons described by the coupled nonlinear Schrodinger models, where the unperturbed equation is nonintegrable.

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.

The collocation method is the method for the numerical solution of integral equations and partial and ordinary differential equations. The main idea of this method is to choose a number of points in the domain and a finite-dimensional space of candidate solutions. So, that solution satisfies the governing equation at the collocation points. The current paper involves developing, and a comprehensive, step-by step procedure for applying the collocation method to the numerical solution of free vibration of tapered Euler-Bernoulli beam. In this stusy, it is assumed the beam rested on variable Winkler foundation. The simplicity of this approximation method makes it an ideal candidate for computer implementation. Finally, the numerical examples are introduced to show efficiency and applicability of quintic B-spline collocation method. Numerical results are demonstrated that quintic B-spline collocation method is very competitive for numerical solution of frequency analysis of tapered beam on variable elastic foundation.

In this study, numerical solution of generalized Rosenau–Kawahara equation with quintic B-spline collocation finite element method has been obtained. First, the generalized Rosenau–Kawahara equation is converted into a coupled differential equation system by the change of variable for the derivative with respect to space variable. Then, the numerical integrations of the resulting system according to time and space were obtained using the Crank–Nicolson-type formulation and quintic B-spline functions, respectively. The obtained numerical scheme has been applied to four model problems. It is seen that the results obtained from the presented scheme are compatible with the analytical solution, the error norms are smaller than those given in the literature, and conservation constants remain virtually unchanged.

In the paper we deal with a quintic Liénard system of the form x ̇ = y − ( a 1 x + a 2 x 3 + a 3 x 5 ) , y ̇ = b 1 x + b 2 x 3 with a Z 2 -symmetry, where a 1 , a 2 , b 1 ∈ R and a 3 b 2 ≠ 0. A complete study of this system with b 2 > 0, called the saddle case, is finished, showing that the system exhibits at most two limit cycles, and the necessary and sufficient conditions are obtained on the existence of two limit cycles and a two-saddle heteroclinic loop. We also present a global bifurcation diagram and the corresponding phase portraits of this system, including Hopf bifurcation, Bautin bifurcation, two-saddle heteroclinic loop bifurcation and double limit cycle bifurcation.

In this paper, we present a Crank-Nicolson collocation method based on modified Quintic B-splines for parabolic PDEs. The time-dependent convection–diffusion equation (CDE) and Burgers’ equation are considered. The integration of the problem is handled by using a modified quintic B-spline, over Crank-Nicolson (C-N) scheme, in space. The time-dependent terms are discretized using FDM. The efficiency as well as the accuracy of the method checked through the six numerical examples. The approximate results are presented and compared with the known exact and other numerical solutions. The Von Neumann stability is also performed.

The aim of this study is to obtain numerical solutions of the modified Fornberg Whitham equation via collocation finite element method combined with operator splitting method. The splitting method is used to convert the original equation into two sub equations including linear and nonlinear part of the equation as a slight modification of splitting idea. After splitting progress, collocation method is used to reduce the sub equations into algebraic equation systems. For this purpose, quintic B-spline base functions are used as a polynomial approximation for the solution. The effectiveness and efficiency of the method and accuracy of the results are measured with the error norms $L_{2}$ and $L_{\infty}$. The presentations of the numerical results are shown by graphics as well.

In this study, we have modified quintic B-spline base function to use for numerical solution of the Burgers’ equation. For this purpose, the numerical algorithm differential quadrature method and Crank–Nicolson scheme are used for space and time discretization, respectively. To deal with nonlinear terms, they are linearized by using Rubin–Graves technique. To observe the accuracy of the present modification, ten different and important test problems are solved successfully. The present results are compared with earlier single methods and mixed methods to see the efficiency of the present algorithm. All comparisons display that the present approach developed the numerical results according to earlier ones. Numerical simulations are plotted at selected time steps to observe the physical behaviour of the waves in 2D and 3D. The rate of convergence related to space is calculated for different examples and reported with error norms.

Recently, zeroing dynamics (ZD) method has shown wonderful effect in solving time-varying problems. Generally, the continuous-time ZD model should be discretized as a discrete-time algorithm for numerical computation. A good choice is to apply Zhang et al. discretization (ZeaD) formulas, which are effective time-discretization formulas. Actually, ZeaD formulas can also be applied to obtaining various discrete-time algorithms, not only the ZD algorithms. In the previous work, some piecemeal ZeaD formulas have been proposed and investigated. However, the relationship between the time-instant number and precision of ZeaD formulas is not found, which is emphatically investigated in this paper. Specifically, the ZeaD formulas from two to nine instants are investigated, and the general ZeaD formula groups are studied. Two-instant and three-instant ZeaD formula groups have linear precision at most. Four-instant ZeaD formula group has quadratic precision at most. Five-instant and six-instant ZeaD formula groups have cubic precision at most. Seven-instant and eight-instant ZeaD formula groups have quartic precision at most. Nine-instant ZeaD formula group has quintic precision at most. Theoretical analyses are presented to substantiate the relationship. Moreover, the ZeaD formulas as well as ZD method are applied to solving time-varying quadratic optimization problem, and the numerical results verify the effectiveness of ZeaD formulas.

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

This is the first part of our project toward proving the Bershadsky–Cecotti–Ooguri–Vafa Feynman graph sum formula of all genera Gromov–Witten invariants of quintic Calabi–Yau threefolds. We introduce the notion of N–mixed-spin-P fields, construct their moduli spaces, their virtual cycles and their virtual localization formulas, and obtain a vanishing result associated with irregular graphs.

In this manuscript, two recent numerical schemes (the trigonometric quintic and exponential cubic B-spline schemes) are employed for evaluating the approximate solutions of the nonlinear Klein-Gordon-Zakharov model. This model describes the interaction between the Langmuir wave and the ion-acoustic wave in a high-frequency plasma. The initial and boundary conditions are constructed via a novel general computational scheme. [ 1 ] has used five different numerical schemes, such as the Adomian decomposition method, Elkalla-expansion method, three-member of the well-known cubic B-spline schemes. Consequently, the comparison between our solutions and that have been given in [ 1 ], shows the accuracy of seven recent numerical schemes along with the considered model. The obtained numerical solutions are sketched in two dimensional and column distribution to explain the matching between the computational and numerical simulation. The novelty, originality, and accuracy of this research paper are explained by comparing the obtained numerical solutions with the previously obtained solutions.

In this paper, we investigate a numerical scheme for the singular Emden–Fowler problem using quintic B-spline. We eliminate the singularity of the problem according to Lopita's law and obtain a new...

We investigate the dromion-like excitations corresponding to intramolecular chain-like proteins. In the present work, the dromion-like excitations are described by using cubic-quintic nonlinear Schrodinger equation (CQNSE) governing the dynamics of proteins and we analytically analyze the velocity (v) of dromion-like structure compared with velocity ( $$v_a$$ ) of acoustical sound waves corresponding to the longitudinal vibrations of protein molecules. Our work is motivated by the effectiveness and powerful mathematical techniques such as modified extended tanh function method and sine–cosine function method for solving CQNSE to obtain dromion-like structures.

One of the problems on which a great deal of focus is being placed today, is how to teach Calculus in the presence of the massive diffusion of Computer Algebra tools and online resources among students. The essence of the problem lies in the fact that, during the problem solving activities, almost all undergraduates can be exposed to certain ''new'' functions, not typically treated at their level. This, without being prepared to handle them or, in some cases, even knowing the meaning of the answer provided by the computer system used. One of these functions is Lambert’s \(W\) function, undoubtedly due to the elementary nature of its definition. In this article we introduce \(W\), in a way that is easy to grasp for first year undergraduate students and we provide some general results concerning polynomial-exponential and polynomial-logarithmic equations. Among the many possible examples of its applications, we will see how \(W\) comes into play in epidemiology in the SIR model. In the second part, using more advanced concepts, we motivate the importance of the Implicit Function Theorem, using it to obtain the power series expansion of the Lambert function around the origin. Based on this approach, we therefore also provide a way to obtain the power series expansion of the inverse of a given smooth function \(f(y)\), when it is assumed that \(f(0)=0,\,f'(0)\neq0\), aided by the computational power of Mathematica\(_\circledR\). Basically, in this way, we present an alternative approach to the Lagrange Bürman Inversion Theorem, although in a particular but relevant case, since the general approach is not at an undergraduate level. A number of good references are [1, pp. 23-28] and [2], where the Lambert function is applied. Finally, these skills are used to take into consideration the particular quintic equation in the unknown \(y\) presented by F. Beukers [3]. Namely, we consider \(x(1+y)^5-y=0\) as an example of an equation for which the power series representation of one of its real solutions is known, calculating, with the same method used for the Lambert function, the first terms of its power series representation.

Beams supported by elastic foundations one of the complex soil-structure interaction problems and analysis carried out using the concept of “beam on elastic foundation” approach. The structural foundation and the soil continuum must act together to support the loads. Developing more realistic foundation models and simplified methods are very important for safe and economical design of such type of structure. In the present study, a first order continuity three nodded beam based on Euler–Bernoulli assumptions and a workable approach for the analysis of beams on Pasternak foundation is attempted. A Matlab code is developed for present formulation. The results, thus obtained, are compared with similar studies done by other researchers, which show very good conformity. Parametric studies are carried out to obtain response for different loading conditions, boundary conditions and foundation parameter.It is concluded that the present formulation has rapid convergence regardless of boundary conditions, aspect ratio and foundation parameters. It behaves extremely well for Euler–Bernoulli beams effectively and efficiently. The effect of the soil coefficient on the response of beams on two parameter elastic foundation is generally larger than that of beam physical and material property.

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.

Our main purpose in this work is to investigate a new solution that represents a numerical behavior for one well-known nonlinear wave equation, which describes the Bona–Smith family of Boussinesq type. A numerical solution has been obtained according to the quintic B-spline collocation method. The method is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The stability of the proposed method has been discussed and presented to be unconditionally stable. The efficiency of the proposed method has been demonstrated by studying a solitary wave motion and interaction of two and three solitary waves. The results are found to be in good agreement with the analytic solution of the system. We demonstrated the physical interpretation of some obtained results graphically with symbolic computation.

Pulp washing is a prime activity in the process industry that involves diffusion–dispersion phenomena. A huge amount of cost, time, and ecological issues are entailed in waste-water management. To reduce this environmental load and to achieve higher efficiency, the mathematical models are developed and solved with different techniques by the various researchers. In the present study, quintic Hermite interpolating polynomials are used to approximate the trial function for solving the mathematical model of diffusion–dispersion phenomena. The purpose behind this study is to derive an accurate result with less CPU time and effect for some important parameters such as Peclet number, cake thickness, and interstitial velocity of the pulp washing process. Two problems, first with the constant coefficient and second with the variable coefficient are worked out by the proposed scheme. After getting the desired results for the linear model, the method is applied to the nonlinear model. The results indicate that the Peclet number plays a leading role in the pulp washing process whereas, the cake thickness and interstitial velocity both are having a lesser effect. The efficiency, accuracy, and applicability of the method is derived using $$\left\| L \right\|_{2}$$ norm, $$\left\| L \right\|_{\infty }$$ norms, and rate of convergence. The suitability of the proposed technique is well weighed up when compared with the earlier published results and displays a wider scope of industrial applicability.

In the light of the potential applications in engineering, electronics, physics, chemistry, and biology, the current work applies several techniques to achieve analytic approximate and numerical solutions of the cubic–quintic Duffing–van der Pol equation. This equation represents a second-order ordinary differential equation with quintic nonlinearity and includes two external periodic forcing terms. A classical approximate solution involves the secular terms is obtained. Unfortunately, this traditional method does not enable us to ignore these secular terms. Additionally, along with the concept of the expanded frequency, a bounded approximate solution is achieved. The Homotopy perturbation method is utilized to obtain an approximate solution with an artificial frequency of the given system. Near the equilibrium points, in the case of the autonomous system, the linearized stability is accomplished. Furthermore, in the case of the non-autonomous system, by means of the multiple time scales, the stability analysis is effectuated, together with the resonance and the non-resonance cases. Numerical computations are performed to demonstrate, graphically, the perturbed solutions as well as the stability/instability regions. Various numerical solutions to initial–boundary value problems are deduced via a three-step finite difference scheme. These are plotted and discussed to show the chaotic nature of solutions.

We investigate the inhomogeneous higher-order nonlinear Schrodinger (INHLS) equation including cubic–quintic–septic (CQS) nonlinear terms and gain or loss with variable coefficients. The exact analytic solution that describes dark soliton-type pulse propagation is found for the model by employing the ansatz method. Unlike the traditional $$\text {tanh}$$ dark soliton in Kerr-type media, the functional form of this novel dark-managed soliton structure takes a $$\text {sech}^{2/3}$$ profile. Parametric conditions are presented in which these optical solitons exist. We also investigated the stability of these dark-managed solitons under some initial perturbations by employing the numerical simulation methods. Finally, the interaction dynamics of two and three dark-managed solitons has been numerically explored.

In the article, we prove the large data scattering for two models, i.e. the defocusing quintic nonlinear Schrodinger equation on R 2 × T and the defocusing cubic nonlinear Schrodinger equation on R 3 × T . Both of the two equations are mass supercritical and energy critical. The main ingredients of the proofs contain global Stricharz estimate, profile decomposition and energy induction method. This paper is the second project of our series work (two papers, together with [31] ) on large data scattering for the defocusing critical NLS with integer index nonlinearity on low dimensional waveguides. At this point, this category of problems are almost solved except for two remaining resonant system conjectures and the quintic NLS problem on R × T .

This paper describes the exceptionally precise results of 6 th -order and 8 th -order nonlinear boundary-value problems(BVPs). Cubic-Nonpolynomial spline(CNPS) and Cubic-polynomial spline(CNPS) are utilized to solve such types of BVPs. We develop the class of numerical techniques for a particular selection of the factors that are associated with nonpolynomial and polynomial splines. Using the developed class of numerical techniques, the problem is reduced to a new system that consists of 2 nd -order BVPs only. The end conditions associated with the BVPs are determined. For each problem, the results obtained by CNPS and CPS is compared with the exact solution. The absolute error(AE) for every iteration is calculated. To show that the suitable responses established by using CNPS and CPS have a higher level of preciseness, the absolute errors of the CNPS and CPS have been compared with different techniques such as DTM, ADM, Parametric septic splines, Variational-iteration method(VIM), Daftardar Jafari strategy, MDM, Cubic B-Spline, Homotopy method(HM), Quintic and Sextic B-spline and observed to be more accurate. Graphs that describe the graphical comparison of CNPS and CPS at n = 10 are also included in this paper.

We prove a vanishing property of the normal form transformation of the 1D cubic nonlinear Schrodinger (NLS) equation with periodic boundary conditions on [0, L]. We apply this property to quintic resonance interactions and obtain a description of dynamics for time up to $$T=\frac{L^2}{\epsilon ^4}$$ , if L is sufficiently large and size of initial data $$\epsilon $$ is small enough. Since T is the characteristic time of wave turbulence, this result implies the absence of wave turbulence behavior of 1D cubic NLS. Our approach can be adapted to other integrable systems without too many difficulties. In the proof, we develop a correspondence between Feynman diagrams and terms in normal forms, which allows us to calculate the coefficients inductively.

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%.

In this work, we deal with the region control of $$C^2$$ interpolation curves and surfaces using a class of rational interpolation splines in one and two dimensions. Simple sufficient data-dependent constraints are derived on the local control parameters to generate $$C^2$$ interpolation curves lying strictly between two given piecewise linear curves and $$C^2$$ interpolation surfaces lying strictly between two given piecewise bi-liner blending quintic interpolation surfaces. Moreover, we also develop an algorithm concerning the application of the $$C^2$$ rational interpolation spline surfaces on image interpolation.

A novel higher order meshing around the airfoil design by using finite simple triangle elements has been presented in this paper. An automated meshing has been performed by linear to sextic order (n = 1, 2, 3, 4, 5 and 6) simple triangle elements using MATLAB code. The subparametric transformation approach has been implemented in the present accurate and efficient meshing. Linear, quadratic, cubic, quartic, quintic and sextic order triangle shaped elements are been utilized to discretize the area near the airfoil shape. An itemized extraction of the edge information, nodal coordinates, the total number of discretized elements and the triangle indices has been carried out from the present meshing. A NACA0018 airfoil design of 18% thickness of max camber is under consideration for the present work. Meshing is executed with the MATLAB code AirfoilHOmesh2d .m by implementing simple changes based on the present 4-digit NACA0018 airfoil profile. The accuracy of an eminent linear mesh generator scheme presented by Gilbert Strang and Persson has been proved and verified. Present work has extended these eminent linear meshing scheme by utilizing subparametric transformation on higher order triangular elements. The algorithm of meshing with higher order has been explained in detail and represented in flowchart form in the Appendix. The data extracted from the meshing can offer a simplified approach to optimize the airfoil shape, for the study of flow around the airfoil in fluid dynamics and the analysis of adverse effects on its external surface in aerospace applications. The advanced higher order meshing is beneficial for the optimization of any symmetric airfoil resulting a better economical fuel usage in the aircraft.

The numerical solutions of the two different forms of the modified Kawahara equation namely bell-shaped soliton solutions and travelling wave solutions that occur thereby the different form of the KdV equation have been investigated. To improve the numerical solutions, two efficient methods have been used together. Firstly, Crank–Nicolson discretization algorithm for time integration is used and then fifth-order quintic B-spline based differential quadrature method for space integration is used. To observe the performance of the present algorithm bell-shaped soliton solution and travelling wave solutions are surveyed. The error norms L 2 and L ∞ are obtained quite less than earlier papers. The invariants and relative changes of invariants are added to sympathize with superior present results.

Certain hybrid prototypes of dispersive optical solitons that we are looking for can correspond to new or future behaviors, observable or not, developed or will be developed by optical media that present the cubic-quintic-septic law coupled, with strong dispersions. The equation considered for this purpose is that of non-linear Schrodinger. The solutions are obtained using the Bogning-Djeumen Tchaho-Kofane method extended to the new implicit Bogning’ functions. Some of the obtained solutions show that their existence is due only to the Kerr law nonlinearity presence. Graphical representations plotted have confirmed the hybrid and multi-form character of the obtained dispersive optical solitons. We believe that a good understanding of the hybrid dispersive optical solitons highlighted in the context of this work allows to grasp the physical description of systems whose dynamics are governed by nonlinear Schrodinger equation as studied in this work, allowing thereby a relevant improvement of complex problems encountered in particular in nonliear optaics and in optical fibers.

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.

The exact solutions of both the cubic Duffing equation and cubic-quintic Duffing equation are presented by using only leaf functions. In previous studies, exact solutions of the cubic Duffing equation have been proposed using functions that integrate leaf functions in the phase of trigonometric functions. Because they are not simple, the procedures for transforming the exact solutions are complicated and not convenient. The first derivative of the leaf function can be derived as the root. This derivative can be factored. These factors or multiplications of factors are exact solutions to the Duffing equation. Some of these exact solutions are of the same type as the cubic Duffing equation reported in previously. Some of these exact solutions satisfy the exact solutions of the cubic--quintic Duffing equations with high nonlinearity. In this study, the relationship between the parameters of these exact solutions and the coefficients of the terms of the Duffing equation is clarified. We numerically analyze these exact solutions, plot the waveform, and discuss the periodicity and amplitude of the waveform., 55 pages, 60 figures, 7 tables

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...

We provide a definition of Vafa-Witten invariants for projective surface Deligne-Mumford stacks, generalizing the construction of Tanaka-Thomas on the Vafa-Witten invariants for projective surfaces inspired by the S-duality conjecture. We give calculations for a root stack over a general type quintic surface, and quintic surfaces with ADE singularities. The relationship between the Vafa-Witten invariants of quintic surfaces with ADE singularities and the Vafa-Witten invariants of their crepant resolutions is also discussed.

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 used an optimal control method involving covariant control equations as optimality conditions, to command the actuators of robot manipulators. These form a coupled system of second order nonlinear ordinary differential equations when associated with the robot motion equations. By solving this system, the control action required to take the robot from an initial to a final state is optimized in a prescribed time. However, the target set of equations exhibited stiffness. Therefore, an adequate solution could only be found for short trajectory durations with readily available numerical methods. We examined a time discretization procedure based on cubic and quintic Hermite finite elements which exhibited superconvergence properties for interpolation. This motivated us to develop a time integration algorithm based on Hermite’s technique, where motion and control equations were perturbed to solve the optimal control problem. The optimal motion of a robotic manipulator was simulated using this algorithm. Our method was compared with a commercial differential equations solver on the basis of specific indicators. It outperformed the commercial solver by effectively solving the stiff set of equations for longer trajectory durations, with the cubic elements performing better than the quintic ones in this sense. The convergence analysis of our method confirmed that the quintic elements are more precise at the cost of increased computational burden, but converge at a lower rate than expected. Controlled motion experiments on a robotic manipulator validated our methodology. Trajectories were smoothly tracked and results exposed further methodology improvements.