Explosive Instability due to four-wave mixing

It is known that an explosive instability can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. The purpose of this Letter is to show that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing: four resonantly interacting wave trains gain energy from a background, and all blowup in a finite time. Unlike singularities associated with self-focussing, these singularities can occur with no spatial structure-the waves blowup everywhere in space simultaneously. We have not yet investigated the effect of spatial structure on a 4-wave explosive instability.     

Electrostatic interactions in molecular dynamics simulation of a three-dimensional system with periodicity in one direction

The Mellin transform and Poisson summation formula are used to derive an expression for the Coulomb interaction energy of a three-dimensional system with periodicity in one direction. Initially, calculations are performed for interactions characterized by any inverse power and, using the analytical continuation of the energy function, one obtains the final expression for the interaction energy of charges. We consider also a special case when two different charges are located on a line parallel to the periodicity direction. The energy and force expressions are identical to those obtained from the Lekner summation which is simply a sum over reciprocal lattice terms. The convergence behaviour of the Lekner summation is compared with that based on the Ewald type approach.     

Summation methods of Coulomb interactions in computer simulations of a system with one—dimensional periodic boundary conditions

The Ewald-type method, its modified version and the Lekner-type method for summing Coulomb interactions in a system periodic along one direction are presented and compared. Advantages and disadvantages of these methods are discussed, and the methods are tested in molecular dynamics simulations of acetone molecules confined to cylindrical silica pores.     

Coulomb interactions in a computer simulation of a system periodic in two directions

The integral representation of the gamma function and the Poisson summation formula are used to calculate the interaction energy of charged particles in a 3-dimensional system periodic in two directions. A parallelogram shape simulation box is considered. Calculations are carried out for interactions described by any inverse power, and analytical continuation of the energy function leads to the final expression for the Coulomb interaction energy. Summation over the simulation box replica along one or the other side of the box base is replaced by summation in reciprocal space. Therefore there are two equivalent formulas for the potential energy that offer the possibility of avoiding slowly convergent series. The energy expressions are identical to those obtained from the Lekner method. The special case is considered where the functions defining the energy are infinite, i.e. when two charges lie on a line parallel to the simulation box side that was chosen to convert real space summation into reciprocal space.