The modulational instability of the higher-order nonlinear Schrödinger equation with fourth-order dispersion and quintic nonlinear terms, describing the propagation of extremely short pulses, is investigated. Several types of gains by modulational instability are shown to exist in both the anomalous and normal dispersion regimes depending on the sign and strength of the higher-order nonlinear t...

By implementing the exact density matrix for the rotating anisotropic harmonic trap, we derive a class of very fast and accurate fourth-order algorithms for evolving the Gross-Pitaevskii equation in imaginary time. Such fourth-order algorithms are possible only with the use of forward, positive time step factorization schemes. These fourth-order algorithms converge at time-step sizes an order-o...

In this paper, a variable-order nonlinear cable equation is considered. The numerical method with first order temporal accuracy and fourth order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second order temporal accuracy and fourth order spatial accuracy. Finally, the resul...

M-eigenvalues of fourth-order partially symmetric tensors play an important role in many real fields such as quantum entanglement and nonlinear elastic materials analysis. In this paper, we give two bounds for the maximal absolute value of all the M-eigenvalues (called the M-spectral radius) of a fourth-order partially symmetric tensor and discuss the relation of them. A numerical example is gi...

A family of fourth and second-order accurate numerical schemes is presented for the solution of nonlinear fourth-order boundary-value problems (BVPs) with two-point boundary conditions. Non-polynomial quintic spline functions are applied to construct the numerical algorithms. This approach generalizes nonpolynomial spline algorithms and provides a solution at every point of the range of integra...

In this paper, we present a new one-parameter fourth-order family of iterative methods for solving nonlinear equations. The new family requires two evaluations of the given function and one of its first derivative per iteration. The well-known Traub–Ostrowski’s fourth-order method is shown to be part of the family. Several numerical examples are given to illustrate the efficiency and performanc...

In this paper we shall reduce the problem of the oscillation of all solutions of certain nonlinear fourth-order dynamic equations to the problem of oscillation of two second-order dynamic equations, which are discussed intensively in the literature. Further oscillation criteria of fourth-order equations are given and proved using integration and Taylor’s formula on time scales. Some conditions ...

The construction of optimal fourth-order iterative schemes for solving univariate nonlinear equations is discussed. Per iteration, the methods consist of three evaluations of the function and they are free from any derivative calculation which property is so fruitful in engineering problems. We analytically show the fourth-order convergence. Numerical examples are considered to confirm the appl...

In this paper, we present fourth-order families of new variants of King’s fourth-order family of methods for solving nonlinear equations. Per iteration the obtained methods require two evaluations of the given function and one of its first derivative. The classical Traub–Ostrowski method is obtained as a special variant of King’s method. Several numerical examples are given to illustrate the ef...

In this paper new fourth order optimal root-finding methods for solving nonlinear equations are proposed. The classical Jarratt’s family of fourth-order methods are obtained as special cases. We then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three. The basins of attraction of existing optimal metho...