Showing papers on "Quintic function published in 2016"
TL;DR: In this article, the authors proved almost sure global well-posedness of the energy-critical defocusing quintic nonlinear wave equation on R 3 with random initial data in H s (R 3 ) × H s − 1 ( R 3 ) for s > 1 2.
76 citations
TL;DR: Differential quadrature methods based on B-spline functions of degree four and five have been introduced to solve advection-diffusion equation numerically and stability analysis for both methods is studied by the use of matrix stability.
45 citations
TL;DR: The problem of constructing a plane polynomial curve with given end points and end tangents, and a specified arc length, and an algorithm to construct interpolants to planar G1 Hermite data, with exact prescribed arc lengths is presented.
42 citations
TL;DR: In this article, the existence of a compact global attractor for the solution semigroup of the dissipative wave equation with a critical quintic nonlinearity in smooth bounded three-dimensional domain is considered based on the recent extension of the Strichartz estimates to the case of bounded domains.
Abstract: The dissipative wave equation with a critical quintic non-linearity in smooth bounded three-dimensional domain is considered Based on the recent extension of the Strichartz estimates to the case of bounded domains, the existence of a compact global attractor for the solution semigroup of this equation is established Moreover, the smoothness of the obtained attractor is also shown
40 citations
TL;DR: In this paper, a nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, unforced cubic-quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution.
Abstract: The nonlinear differential equation governing the periodic motion of the one-dimensional, undamped, unforced cubic–quintic Duffing oscillator is solved exactly, providing exact expressions for the period and the solution. The period is given in terms of the complete elliptic integral of the first kind and the solution involves Jacobi elliptic functions. Some particular cases obtained varying the parameters that characterize this oscillator are presented and discussed. The behaviour of the period as a function of the initial amplitude is analysed, the exact solutions and velocities for several values of the initial amplitude are plotted, and the Fourier series expansions for the exact solutions are also obtained. All this allows us to conclude that the quintic term appearing in the cubic–quintic Duffing equation makes this nonlinear oscillator not only more complex but also more interesting to study.
33 citations
TL;DR: The lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices as discussed by the authors, where the lattice shape of the ring of the integers is ordered by their absolute discriminants.
Abstract: For are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.
31 citations
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TL;DR: In this paper, the localization formula is derived, and algorithms toward evaluating these Gromov-Witten invariants are derived for the quintic Calabi-Yau threefold.
Abstract: This is the second part of the project toward an effective algorithm to evaluate all genus Gromov-Witten invariants of quintic Calabi-Yau threefolds. In this paper, the localization formula is derived, and algorithms toward evaluating these Gromov-Witten invariants are derived.
29 citations
TL;DR: In this article, it was shown that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation have finite rank, which is a special case of a conjecture by Broue, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group.
22 citations
01 Aug 2016
TL;DR: An interpolation method based on quintic non-uniform rational B-spline(NURBS) is proposed to construct curves when to plan the trajectory of manipulators with respect to three objectives, time optimal, energy optimal and smoothness optimal, and shows that quintic NURBS curve can get high-order continuous trajectories and NSGA-II algorithm can provide an effective approach to get the perfect Pareto solutions.
Abstract: In this paper an interpolation method based on quintic non-uniform rational B-spline(NURBS) is proposed to construct curves when to plan the trajectory of manipulators with respect to three objectives, time optimal, energy optimal and smoothness optimal. The mathematical model of quintic NURBS curve is set up to gain high-order continuous trajectories with endpoint configuration parameters can be specified, and a fast and elitist multi-objective genetic algorithm (NSGA-II) is adopted to optimize the trajectory of manipulators aims to get a series of Pareto optimal solutions under the three objectives. Through the simulation with six-degree of freedom robot shows that quintic NURBS curve can get high-order continuous trajectories and NSGA-II algorithm can provide an effective approach to get the perfect Pareto solutions for quintic NURBS curve. By constructing a average fuzzy membership function, a potential optimal solution can be selected from Pareto optimal set, then the high-order continuous optimal trajectory can be obtained.
21 citations
TL;DR: In this article, Fisher-Kolmogorov equation is solved numerically by adopting a differential quadrature technique that uses quintic B-spline as the basis functions for space integration.
Abstract: In the present manuscript, Fisher-Kolmogorov equation is solved numerically by adopting a differential quadrature technique that uses quintic B-spline as the basis functions for space integration. The derivatives are approximated using differential quadrature method. The weighting coefficients are obtained by semi-explicit algorithm. Five-band Thomas algorithm has been employed to solve the resultant algebraic system that can be reduced into a penta-diagonal matrix. Stability analysis of method has also been done. The accuracy of the proposed scheme is demonstrated by applying on three test problems. Theoretical attributes such as existence, uniqueness and regularity of Fisher-Kolmogorov equations are also conferred. The outcomes are depicted graphically to confirm accuracy of the findings and performance of this method and a comparative study is done with results available in the literature. The computed results are found to be in good agreement with the analytical solutions.
21 citations
TL;DR: In this article, a general dispersion relation is derived and evaluated by using a phase diagram analysis, focusing on the conditions of existence and solvability of the dispersion relations.
Abstract: TE-polarized electromagnetic waves, guided by a three-layer slab structure consisting of a central film with quartic permittivity placed between two half spaces with Kerr permittivity, are studied. Traveling-wave solutions of Maxwell's equations are expressed in terms of Weierstrass's elliptic function $\ensuremath{\wp}$. A general dispersion relation is derived and evaluated by using a phase diagram analysis. Emphasis is placed on the conditions of existence and solvability of the dispersion relation. Numerical results are presented.
TL;DR: In this article, a numerical method is proposed for the numerical solution of a coupled system of KdV (CKdV) equation with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points.
Abstract: In the present paper, a numerical method is proposed for the numerical solution of a coupled system of KdV (CKdV) equation with appropriate initial and boundary conditions by using collocation method with quintic B-spline on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms, are computed. Three invariants of motion are predestined to determine the preservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is easy to apply. We make linearization for the nonlinear term.
TL;DR: A one-dimensional model of the parity-time ( PT)-symmetric coupler is introduced, with mutually balanced linear gain and loss acting in the two cores, and nonlinearity represented by the combination of self-focusing cubic and defocusing quintic terms in each core.
Abstract: We introduce a one-dimensional model of the parity-time ( PT)-symmetric coupler, with mutually balanced linear gain and loss acting in the two cores, and nonlinearity represented by the combination of self-focusing cubic and defocusing quintic terms in each core. The system may be realized in optical waveguides, in the spatial and temporal domains alike. Stationary solutions for PT-symmetric solitons in the systems are tantamount to their counterparts in the ordinary coupler with the cubic-quintic nonlinearity, where the spontaneous symmetry breaking of solitons is accounted for by bifurcation loops. A novel problem is stability of the PT-symmetric solitons, which is affected by the competition of the PT symmetry, linear coupling, cubic self-focusing, and quintic defocusing. As a result, the solitons become unstable against symmetry breaking with the increase of the energy (alias integral power, in terms of the spatial-domain realization), and they retrieve the stability at still larger energies. Above a certain value of the strength of the quintic self-defocusing, the PT symmetry of the solitons becomes unbreakable. In the same system, PT-antisymmetric solitons are entirely unstable. We identify basic scenarios of the evolution of unstable solitons, which may lead to generation of additional ones, while stronger instability creates expanding quasi-turbulent patterns with limited amplitudes. Collisions between stable solitons are demonstrated to be quasi-elastic.
TL;DR: In this paper, the authors studied the propagation of hydrodynamic wave packets and media with negative refractive index in a quintic derivative nonlinear Schrodinger (DNLS) equation.
Abstract: The propagation of hydrodynamic wave packets and media with negative refractive index is studied in a quintic derivative nonlinear Schrodinger (DNLS) equation. The quintic DNLS equation describe the wave propagation on a discrete electrical transmission line. We obtain a Lagrangian and the invariant variational principle for quintic DNLS equation. By using a class of ordinary differential equation, we found four types of exact solutions of the quintic DNLS equation, which are kink-type solitary wave solution, antikink-type solitary wave solution, sinusoidal solitary wave solution, bell-type solitary wave solution. By applying the modulation instability to discuss stability analysis of the obtained solutions. Modulation instabilities of continuous waves and localized solutions on a zero background have been investigated.
TL;DR: In this paper, the one-dimensional nonlinear Schrodinger equation with the cubic-quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function, is studied.
Abstract: We study the one-dimensional nonlinear Schrodinger equation with the cubic–quintic combination of attractive and repulsive nonlinearities, and a trapping potential represented by a delta-function. We determine all bound states with a positive soliton profile through explicit formulas and, using bifurcation theory, we describe their behavior with respect to the propagation constant. This information is used to prove their stability by means of the rigorous theory of orbital stability of Hamiltonian systems. The presence of the trapping potential gives rise to a regime where two stable bound states coexist, with different powers and same propagation constant.
TL;DR: In this article, a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic was shown to be equivalent to the Gromov-Witten theory of the Fermat quintic polynomial.
Abstract: We prove a version of the Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic. In particular we calculate the genus-zero FJRW theory for the pair (W, G) where W is the Fermat quintic polynomial and G = SL(W). We identify it with the Gromov-Witten theory of the mirror quintic three-fold via an explicit analytic continuation and symplectic transformation. In the process we prove a mirror theorem for the corresponding Landau-Ginzburg model (W,G).
01 Jan 2016
TL;DR: In this paper, a linear stability analysis based on von Neu-mann approximation theory of the numerical scheme is investigated to demonstrate the precise and efficiency of the proposed method, the motion of solitary wave is studied by calculating the error norms L2 and L1.
Abstract: In this paper, the Rosenau-KdV equation that is one of the significant equations in physics was discussed. The collo- cation finite element method is implemented to find the numerical simulation of the dispersive shallow water waves with Rosenau-KdV equation using the quintic B-spline basis functions. A linear stability analysis based on von Neu- mann approximation theory of the numerical scheme is investigated. To demonstrate the precise and efficiency of the proposed method, the motion of solitary wave is studied by calculating the error norms L2 and L1. The invariants I1, I2 and their relative changes have been computed to define the conservation properties of the simulation. As a result, the obtained results are found better than some recent results. MSC: 35Q51 † 35Q53 † 35Q58
TL;DR: In this paper, the existence of a local analytic first integral for a family of quintic systems with homogeneous nonlinearities has been shown in polynomial families of Lotka-Volterra systems.
TL;DR: In this paper, the generalized quintic complex Ginzburg-Landau equation is considered to be solved by means of the homotopy analysis method (HAM), by plotting the h-curve of the examples, the region of convergence is determined.
TL;DR: In this paper, the quintic B-spline method is applied directly to the solution of the boundary value problems without reducing the order of the problems, and the convergence analysis is shown to have uniform convergence of the second order.
TL;DR: Based on the B–spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed.
Abstract: The problems of determining the B---spline form of a C2 Pythagorean---hodograph (PH) quintic spline curve interpolating given points, and of using this form to make local modifications, are addressed. To achieve the correct order of continuity, a quintic B---spline basis constructed on a knot sequence in which each (interior) knot is of multiplicity 3 is required. C2 quintic bases on uniform triple knots are constructed for both open and closed C2 curves, and are used to derive simple explicit formulae for the B---spline control points of C2 PH quintic spline curves. These B-spline control points are verified, and generalized to the case of non---uniform knots, by applying a knot removal scheme to the Bezier control points of the individual PH quintic spline segments, associated with a set of six---fold knots. Based on the B---spline form, a scheme for the local modification of planar PH quintic splines, in response to a control point displacement, is proposed. Only two contiguous spline segments are modified, but to preserve the PH nature of the modified segments, the continuity between modified and unmodified segments must be relaxed from C2 to C1. A number of computed examples are presented, to compare the shape quality of PH quintic and "ordinary" cubic splines subject to control point modifications.
TL;DR: In this article, it was shown that the Fermat quintic curve with 15 D4-points is projectively equivalent to the dihedral group D4 of order eight, and the number of D4 points for C equals 0, 1, 3, 5, 5 or 15.
Abstract: Let C be a nonsingular plane quintic curve over the complex number field C, and let πP : C → P be a projection from P ∈ C. Let LP be the Galois closure of the field extension C(C)/C(P) induced by πP , where C(C) and C(P) are the rational function fields of C and P, respectively. We call the point P a D4-point if the Galois group of LP /C(P) is isomorphic to the dihedral group D4 of order eight. In this paper, we prove that the number of D4-points for C equals 0, 1, 3, 5, or 15, and show that the curve with 15 D4-points is projectively equivalent to the Fermat quintic curve.
01 Jun 2016
TL;DR: In this article, a numerical method is proposed for the numerical solution of the Hirota equation by using collocation method with the quintic B-spline, which is shown to be unconditionally stable using von-Neumann technique.
Abstract: In the present article, a numerical method is proposed for the numerical solution of the Hirota equation by using collocation method with the quintic B-spline. The method is shown to be unconditionally stable using von-Neumann technique. To test accuracy the error norms L2, L∞ are computed. Two invariants of motion are predestined to determine the conservation properties of the problem, and the numerical scheme leads to careful and active results. Furthermore, interaction of two and three solitary waves is shown. These results show that the technique introduced here is plain to apply
TL;DR: In this article, the amplitude of the dominant mode is approximated by a stochastic ordinary differential equation with quadratic and cubic nonlinearity, and the effect of additive degenerate noise on the amplitude is investigated.
Abstract: In this paper, we are interested in the approximation of a stochastic generalized Swift-Hohenberg equation with quadratic and cubic nonlinearity by using the natural separation of time-scales near a change of stability. The main results show that the behavior of the SPDE is well approximated by a stochastic ordinary differential equation describing the amplitude of the dominant mode. The cubic and the quadratic nonlinearities lead to cubic nonlinearities of opposite sign. Here we study the interesting case, where both contributions cancel and in the right scaling a quintic nonlinearity emerges in the amplitude equation. Also, we give a brief indication of how the effect of additive degenerate noise (i.e. noise that does not act directly to the dominant mode) might lead to the stabilization of the trivial solution.
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TL;DR: In this paper, on-shell scattering amplitudes involving fermions at the tree level in open superstring field theory were calculated for complete action with a constraint on the Ramond sector and for the covariant formulation developed by Sen with spurious free fields.
Abstract: We calculate on-shell scattering amplitudes involving fermions at the tree level in open superstring field theory. We confirm that four-point and five-point amplitudes in the world-sheet path integral with the standard prescription using picture-changing operators are reproduced. For the four-point amplitudes, we find that the quartic interaction required by gauge invariance adjusts the different assignment of picture-changing operators in the $s$-channel and in the $t$-channel of Feynman diagrams with two cubic vertices. For the five-point amplitudes, the correct amplitudes are reproduced in a more intricate way via the quartic and quintic interactions. Our calculations can be interpreted as those for a complete action with a constraint on the Ramond sector or as those for the covariant formulation developed by Sen with spurious free fields.
TL;DR: In this paper, a method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic-quintic complex Ginzburg-Landau equation about numerically determined stationary solutions is described.
Abstract: We describe a computational method to compute spectra and slowly decaying eigenfunctions of linearizations of the cubic–quintic complex Ginzburg–Landau equation about numerically determined stationary solutions. We compare the results of the method to a formula for an edge bifurcation obtained using the small dissipation perturbation theory of Kapitula and Sandstede. This comparison highlights the importance for analytical studies of perturbed nonlinear wave equations of using a pulse ansatz in which the phase is not constant, but rather depends on the perturbation parameter. In the presence of large dissipative effects, we discover variations in the structure of the spectrum as the dispersion crosses zero that are not predicted by the small dissipation theory. In particular, in the normal dispersion regime we observe a jump in the number of discrete eigenvalues when a pair of real eigenvalues merges with the intersection point of the two branches of the continuous spectrum. Finally, we contrast the method to computational Evans function methods.
TL;DR: An axisymmetric viscous flow, generated by two large parallel plates slowly approaching each other is investigated and the steady nonlinear governing equations are converted into a fourth-order nonlinear differential equation using integrability condition.
Abstract: An axisymmetric viscous flow, generated by two large parallel plates slowly approaching each other is investigated. The steady nonlinear governing equations are converted into a fourth-order nonlinear differential equation using integrability condition. The resulting nonlinear boundary value problem is solved using quintic B-spline collocation and sinc-collocation methods. The approach consists of reducing the problem to a set of algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the techniques and a comparison is made with existing results in the literature.
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TL;DR: In this paper, the analytic continuation of solutions to the hypergeometric differential equation of order $n$ to the third regular singularity, usually denoted $z=1), with the help of recurrences of their Mellin-Barnes integral representations, is performed.
Abstract: We perform the analytic continuation of solutions to the hypergeometric differential equation of order $n$ to the third regular singularity, usually denoted $z=1$, with the help of recurrences of their Mellin--Barnes integral representations. In the resonant case, there are necessarily logarithmic solutions. We apply the result to Picard-Fuchs equations of certain one--parameter families of Calabi--Yau manifolds, known as the mirror quartic and the mirror quintic.
TL;DR: In this article, the modulational instability of plane waves under competing nonlocal cubic-local quintic nonlinearities is investigated analytically and numerically, and the generic properties of the MI gain spectra are then demonstrated for the Gaussian response function, exponential response function and rectangular response function.
Abstract: We investigate analytically and numerically the modulational instability (MI) of plane waves under competing nonlocal cubic-local quintic nonlinearities. The generic properties of the MI gain spectra are then demonstrated for the Gaussian response function, exponential response function, and rectangular response function. Special attention is paid to competing nonlocal cubic-local quintic nonlinearities on the MI. We observe that the focusing local quintic nonlinearity increases the growth rate and bandwidth of instability contrary to the small values of defocusing local quintic nonlinearity which decrease the growth rate and bandwidth of instability. Numerical simulations of the full model equation describing the dynamics of the waves are been carried out and leads to the development of pulse trains, depending upon the sign the quintic nonlinearity.