Quintic function

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

Solitons under spatially localized cubic-quintic-septimal nonlinearities

Abstract: We explore stability regions for solitons in the nonlinear Schrodinger equation with a spatially confined region carrying a combination of self-focusing cubic and septimal terms, with a quintic one of either focusing or defocusing sign. This setting can be implemented in optical waveguides based on colloids of nanoparticles. The solitons stability is identified by solving linearized equations for small perturbations, and is found to fully comply with the Vakhitov-Kolokolov criterion. In the limit case of tight confinement of the nonlinearity, results are obtained in an analytical form, approximating the confinement profile by a delta-function. It is found that the confinement greatly increases the largest total power of stable solitons, in the case when the quintic term is defocusing, which suggests a possibility to create tightly confined high-power light beams guided by the spatial modulation of the local nonlinearity strength.

Watson's method of solving a quintic equation

Abstract: Watson's method for determining the roots of a solvable quintic equation in radical form is examined in complete detail. New methods in the spirit of Watson are constructed to cover those exceptional cases to which Watson's original method does not apply, thereby making Watson's method completely general. Examples illustrating the various cases that arise are presented.

A Computational Solution to a Question by Beauville on the Invariants of the Binary Quintic

Abstract: We obtain an alternate proof of an injectivity result by Beauville for a map from the moduli space of quartic del Pezzo surfaces to the set of conjugacy classes of certain subgroups of the Cremona group. This amounts to showing that a projective configuration of five distinct unordered points on the line can be reconstructed from its five projective four-point subconfigurations. This is done by reduction to a question in the classical invariant theory of the binary quintic, which is solved by computer-assisted methods. More precisely, we show that six specific invariants of degree 24, the construction of which was explained to us by Beauville, generate all invariants the degrees of which are divisible by 48.

Straight Beams Rested on Nonlinear Elastic Foundations – Part 2 (Numerical Solutions, Results and Evaluation)

Abstract: This paper presents numerical solutions of straight plane beam structures rested on an elastic (Winkler's) foundation. It is a continuation of our previous work (see Part 1 of this article) focused on practical applications and solutions including nonlinearities in the foundation (i.e. bilateral linear, bilateral linear + cubic, bilateral linear + cubic + quintic approximations and unilateral approximation for dependencies of reaction forces on deflection in the foundation). For solutions of nonlinear problems of mechanics (i.e. differential 4th-order equations), the Finite Difference Method (i.e. the Central Difference Method) is applied in combination with the Newton (Newton–Raphson) Method. Finally, in one example, linear and nonlinear approaches are solved, evaluated and compared. In some cases, there are evident major differences between the linear and nonlinear solutions.