Two optimized symmetric eight-step implicit methods for initial-value problems with oscillating solutions

In this paper, we present two optimized eight-step symmetric implicit methods with phase-lag order ten and infinite (phase-fitted). The methods are constructed to solve numerically the radial time-independent Schr?dinger equation with the use of the Woods–Saxon potential. They can also be used to integrate related IVPs with oscillating solutions such as orbital problems. We compare the two new methods to some recently constructed optimized methods from the literature. We measure the efficiency of the methods and conclude that the new method with infinite order of phase-lag is the most efficient of all the compared methods and for all the problems solved. T. E. Simos—Highly Cited Researcher, Active Member of the European Academy of Sciences and Arts.     

High order multistep methods with improved phase-lag characteristics for the integration of the Schr?dinger equation

In this work we introduce a new family of 12-step linear multistep methods for the integration of the Schr?dinger equation. The new methods are constructed by adopting a new methodology which improves the phase lag characteristics by vanishing both the phase lag function and its first derivatives at a specific frequency. This results in decreasing the sensitivity of the integration method on the estimated frequency of the problem. The efficiency of the new family of methods is proved via error analysis and numerical applications. T. E. Simos is a highly cited researcher, active member of the European Academy of Sciences and Arts. Corresponding member of the European Academy of Sciences, corresponding member of European Academy of Arts, Sciences and Humanities.     

Author Index

Authors Index

Author Index     

Experiments in quadratic spatial soliton generation and steering in a noncollinear geometry

Generation of spatial solitons with noncollinear excitation beams was experimentally investigated for type II second-harmonic generation in KTP. Spatial switching at a distance of 220microm and steering in a 330-microm range were demonstrated. Changes in soliton behavior induced by modification of the phase-matching condition (and) or by an imbalance of the inputs at the fundamental frequency have been characterized.     

Efficiency of quadratic soliton generation

We report what is believed to be the first experimental demonstration of soliton content as a function of the intensity of input light for multicolor quadratic solitons. Experiments were conducted with spatial solitons excited with Gaussian beams in a bulk crystal cut for second-harmonic generation. The effects introduced by the finite crystal length and by the scheme employed to elucidate the soliton energy are highlighted.     

Validation tests for the innovation distribution in INAR time series models

Goodness-of-fit tests are proposed for the innovation distribution in INAR models. The test statistics incorporate the joint probability generating function of the observations. Special emphasis is given to the INAR(1) model and particular instances of the procedures which involve innovations from the general family of Poisson stopped-sum distributions. A Monte Carlo power study of a bootstrap version of the test statistic is included as well as a real data example. Generalizations of the proposed methods are also discussed.     

An economical two-step method with optimal phase and stability properties for problems in chemistry

A new ECONO2STEP (economical two-step) method with eliminated phase-lag and its derivatives up to order six is developed in this paper. The newly developed method can be applied to initial or boundary value problems with oscillating and/or periodical solutions. We applied the new method to problems in Quantum Chemistry. The newly introduced method is called economical since using the minimum number of function evaluations per step achieves the highest possible algebraic order.

     

New five-stages finite difference pair with optimized phase properties

A new five-stages symmetric two-step finite difference pair is developed, for the first time in the literature, in this paper. This new finite difference pair has optimal phase and stability properties The main characteristics of the new finite difference pair are:

1. 1.
   it is of symmetric type,
    
2. 2.
   it is of two-step algorithm,
    
3. 3.
   it is of five-stages,
    
4. 4.
   it is of twelfth-algebraic order,
    
5. 5.
   the new nonlinear finite difference pair is produced using the following approximations:

   • An approximation developed on the first layer on the point \(x_{n-1}\),
   • An approximation developed on the second layer on the point \(x_{n-1}\),
   • An approximation developed on the third layer on the point \(x_{n-1}\),
   • An approximation developed on the fourth layer on the point \(x_{n}\) and finally,
   • An approximation developed on the fifth (final) layer on the point \(x_{n+1}\),
    
6. 6.
   it has vanished the phase-lag and its first derivative,
    
7. 7.
   it has optimized stability properties for the general problems,
    
8. 8.
   it is a P-stable method since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).
    

A full numerical analysis (error and stability analysis) is given for the new finite difference pair.

The effectiveness of the new finite difference pair is evaluated by applying it on the approximate solution of systems of coupled differential equations of the Schrödinger form.     

A new family of symmetric linear four-step methods for the efficient integration of the Schrödinger equation and related oscillatory problems

In this article we develop a family of three explicit symmetric linear four-step methods. The new methods, with nullified phase-lag, are optimized for the efficient solution of the Schrödinger equation and related oscillatory problems. We perform an analysis of the local truncation error of the methods for the general case and for the special case of the Schrödinger equation, where we show the decrease of the maximum power of the energy in relation to the corresponding classical methods. We also perform a periodicity analysis, where we find that there is a direct relationship between the periodicity intervals of the methods and their local truncation errors. In addition we determine their periodicity regions. We finally compare the new methods to the corresponding classical ones and other known methods from the literature, where we show the high efficiency of the new methods.