Quintic function

About: Quintic function is a research topic. Over the lifetime, 1677 publications have been published within this topic receiving 26780 citations.

Analytical solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrödinger equation with different external potentials

Abstract: A large family of analytical solitary wave solutions to the generalized nonautonomous cubic-quintic nonlinear Schr\"odinger equation with time- and space-dependent distributed coefficients and external potentials are obtained by using a similarity transformation technique. We use the cubic nonlinearity as an independent parameter function, where a simple procedure is established to obtain different classes of potentials and solutions. The solutions exist under certain conditions and impose constraints on the coefficients depicting dispersion, cubic and quintic nonlinearities, and gain (or loss). We investigate the space-quadratic potential, optical lattice potential, flying bird potential, and potential barrier (well). Some interesting periodic solitary wave solutions corresponding to these potentials are then studied. Also, properties of a few solutions and physical applications of interest to the field are discussed. Finally, the stability of the solitary wave solutions under slight disturbance of the constraint conditions and initial perturbation of white noise is discussed numerically; the results reveal that the solitary waves can propagate in a stable way under slight disturbance of the constraint conditions and the initial perturbation of white noise.

One-dimensional gap solitons in quintic and cubic-quintic Fractional nonlinear Schr\"{o}dinger equations with a periodically modulated linear potential

Abstract: Competing nonlinearities, such as the cubic (Kerr) and quintic nonlinear terms whose strengths are of opposite signs (the coefficients in front of the nonlinearities), exist in various physical media (in particular, in optical and matter-wave media). A benign competition between self-focusing cubic and self-defocusing quintic nonlinear nonlinearities (known as cubic-quintic model) plays an important role in creating and stabilizing the self-trapping of D-dimensional localized structures, in the contexts of standard nonlinear Schrodinger equation. We incorporate an external periodic potential (linear lattice) into this model and extend it to the space-fractional scenario that begins to surface in very recent years---the nonlinear fractional Schrodinger equation (NLFSE), therefore obtaining the cubic-quintic or the purely quintic NLFSE, and investigate the propagation and stability properties of self-trapped modes therein. Two types of one-dimensional localized gap modes are found, including the fundamental and dipole-mode gap solitons. Employing the techniques based on the linear-stability analysis and direct numerical simulations, we get the stability regions of all the localized modes; and particularly, the anti-Vakhitov-Kolokolov criterion applies for the stable portions of soliton families generated in the frameworks of quintic-only nonlinearity and competing cubic-quintic nonlinear terms.

Pythagorean-hodograph quintic transition curves of monotone curvature

Abstract: Walton and Meek [Walton, D.J. and Meek, D.S., A Pythagorean-hodograph quintic spiral. Computer Aided Design , 1996, 28 , 943–950] have recently advocated the use of Pythagorean-hodograph quintics of monotone curvature, or “PH spirals” for short, as transitional elements that give G 2 connections of linear and circular arcs in applications such as layout of highways and railways—in which context PH curves provide the important advantage of rational offsets and exact rectifications . They construct a PH quintic, interpolating an initial point and tangent and a final tangent, with monotone curvature variation from zero to a given (extremum) final value. We show that using the complex representation for PH curves greatly simplifies this problem and also reveals that the method of Walton and Meek yields a special instance among a one-parameter family of solutions. The additional degree of freedom of PH spirals identified herein relaxes constraints otherwise required to guarantee their existence, and offers the designer precise control over their total length and/or the ability to fine-tune their curvature distributions.