Applications of extended simple equation method on unstable nonlinear Schrödinger equations

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x
 2,…,x
 p
 ,exp (±wx),xexp (±wx),…,x
 m
 exp (±w
 x)} and the second is based on the exact integration of the functions of the form: {1,x,x
 2,…,x
 p
 ,exp (±wx),exp (±2wx),…,exp (±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrodinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrodinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrodinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrodinger equation. Finally, numerical results for the resonance problem of the radial Schrodinger equation are presented.

Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation

In this paper, we investigate the connection between • closed Newton-Cotes formulae, • trigonometrically-fitted methods, • symplectic integrators and • efficient integration of the Schrodinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. Based on the closed Newton-Cotes formulae, we also develop trigonometrically-fitted symplectic methods. An error analysis for the one-dimensional Schrodinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.

High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrodinger equation

In this work we construct new Runge-Kutta-Nystrom methods especially designed to integrate exactly the test equation y^''=-w^2y. We modify two existing methods: the Runge-Kutta-Nystrom methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrodinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential.

New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation

The solution of the one-dimensional time-independent Schrodinger equation is considered by exponential-fitting symplectic integrators. The Schrodinger equation is first transformed into a Hamiltonian canonical equation. Numerical results are obtained for the one-dimensional harmonic oscillator and the hydrogen atom.

Exponential-fitting symplectic methods for the numerical integration of the Schrödinger equation

In this work we construct a modified trigonometrically fitted symplectic Runge Kutta Nystrom method based on the forth order five stages method of Calvo and Sanz-Serna. We apply the new method on the numerical integration of the two-body problem.

A fourth order modified trigonometrically fitted symplectic Runge-Kutta-Nyström method

In this work we consider Symplectic Runge Kutta Nystrom methods with five stages. A new fourth algebraic order method with phase-lag order eight is presented. Also the symplectic Runge Kutta Nystrom of Calvo and Sanz Serna with five stages and fourth order is modified to produce a phase-fitted method. We apply th e new methods on several Hamiltonian systems and on the computation of the eigenvalues of the Schrodinger Equation.

http://naturalspublishing.com/files/published/7ems77776aj58u.pdf

Symplectic Runge-Kutta-Nystr ¨ om Methods with Phase-Lag Oder 8 and Infinity

In this work we consider Two Derivative Runge-Kutta methods. We present the order conditions up to order six after applying simplifying conditions.

Order conditions for two derivative Runge Kutta methods up to order six

In this work we consider Symplectic Runge Kutta Nystrom (SRKN) methods with minimum phase‐lag. Also phase fitted SRKN methods are considered. We modify an existing SRKN method of Calvo and SanzSerna with five stages and fourth order.

Th. Monovasilis

Papers

Exponential-fitting symplectic methods for the numerical integration of the Schrödinger equation

A fourth order modified trigonometrically fitted symplectic Runge-Kutta-Nyström method

Symplectic Runge-Kutta-Nystr ¨ om Methods with Phase-Lag Oder 8 and Infinity

Order conditions for two derivative Runge Kutta methods up to order six

Symplectic Runge‐Kutta‐Nyström Methods with Phase‐Lag Order Six and Infinity