Solvability of a class of PT-symmetric non-Hermitian Hamiltonians：Bethe ansatz method

We use the Bethe ansatz method to investigate the Schrdinger equation for a class of PT-symmetric non-Hermitian Hamiltonians. Elementary exact solutions for the eigenvalues and the corresponding wave functions are obtained in terms of the roots of a set of algebraic equations. Also, it is shown that the problems possess sl（2） hidden symmetry and then the exact solutions of the problems are obtained by employing the representation theory of sl（2） Lie algebra. It is found that the results of the two methods are the same.

Traveling Wave Solution of the Modified Benjamin-Bona-Mahony Equation

In this paper, the ansatze method is implemented to study the exact solutions for the modified Benjamin-Bona-Mahony equation (mBBM). The singular-shaped traveling wave solution, the Bell-shape is traveling wave solution, the kink-shaped traveling wave solution and the periodic traveling wave solution is obtained. With the assist of computational software MATLAB, the graphical exemplifications of solutions are illustrated of the two-dimension (2D) and three-dimension (3D) plots.

Bright and dark soliton solutions for some nonlinear fractional differential equations

In this work, we propose a new approach, namely ansatz method, for solving fractional differential equations based on a fractional complex transform and apply it to the nonlinear partial space-time fractional modified Benjamin-Bona- Mahoney （mBBM） equation, the time fractional mKdV equation and the nonlinear fractional Zoomeron equation which gives rise to some new exact solutions. The physical parameters in the soliton solutions： amplitude, inverse width, free parameters and velocity are obtained as functions of the dependent model coefficients. This method is suitable and more powerful for solving other kinds of nonlinear fractional PDEs arising in mathematical physics. Since the fractional deriva- tives are described in the modified Riemann-Liouville sense.

Singular and non-topological soliton solutions for nonlinear fractional differential equations

In this article, the fractional derivatives are described in the modified Riemann-Liouville sense. We propose a new approach, namely an ansatz method, for solving fractional differential equations（FDEs） based on a fractional complex transform and apply it to solve nonlinear space-time fractional equations. As a result, the non-topological as well as the singular soliton solutions are obtained. This method can be suitable and more powerful for solving other kinds of nonlinear fractional FDEs arising in mathematical physics.

Explicit solutions to some nonlinear physical models by a two-step ansatz

Explicit solutions are derived for some nonlinear physical model equations by using a delicate way of two-step ansatz method.

ANALYTICAL SOLUTIONS FOR SOME NONLINEAR EVOLUTION EQUATIONS

The following partial differential equations are studied: generalized fifth-order KdV equation,water wave equation, Kupershmidt equation, couples KdV equation. The analytical solutions to these problems via using various ansatzes by introducing a second-order ordinary differential equation are found out.

Optimal analytical and numerical approximations to the(un)forced(un)damped parametric pendulum oscillator

The(un)forced(un)damped parametric pendulum oscillator(PPO)is analyzed analytically and numerically using some simple,effective,and more accurate techniques.In the first technique,the ansatz method is employed for analyzing the unforced damped PPO and for deriving some optimal and accurate analytical approximations in the form of angular Mathieu functions.In the second approach,some approximations to(un)forced damped PPO are obtained in the form of trigonometric functions using the ansatz method.In the third approach,He’s frequency-amplitude principle is applied for deriving some approximations to the(un)damped PPO.In the forth approach,He’s homotopy technique is employed for analyzing the forced(un)damped PPO numerically.In the fifth approach,the p-solution Method,which is constructed based on Krylov–Bogoliúbov Mitropolsky method,is introduced for deriving an approximation to the forced damped PPO.In the final approach,the hybrid Padé-finite difference method is carried out for analyzing the damped PPO numerically.All proposed techniques are compared to the fourth-order Runge–Kutta(RK4)numerical solution.Moreover,the global maximum residual distance error is estimated for checking the accuracy of the obtained approximations.The proposed methodologies and approximations can help many researchers in studying and investigating several nonlinear phenomena related to the oscillations that can arise in various branches of science,e.g.waves and oscillations in plasma physics.