Narrow Results

We tested the parallelization of explicit schemes for the solution of non-linear classical field theories of complex scalar fields which are capable of simulating hadronic collisions. Our attention focused on collisions in a fractional model with a particularly rich inelastic spectrum of final states. Relativistic collisions of all types were performed by computer on large lattices (64 to 256 sites per dimension). The stability and accuracy of the objects were tested by the use of two other methods of solutions: Pseudo-spectral and semi-implicit. Parallelization of the Fortran code on a 64-transputer MIMD Volvox machine revealed, for certain topologies, communication deadlock and less-than-optimum routing strategies when the number of transputers used was less than the maximum. The observed speedup, for N transputers in an appropriate topology, is shown to scale approximately as N, but the overall gain in execution speed, for physically interesting problems, is a modest 2–3 when compared to state-of-the-art workstations.

A detailed description and validation of a recently developed integration scheme is here reported for one- and two-dimensional reaction–diffusion models. As paradigmatic examples of this class of partial differential equations the complex Ginzburg–Landau and the Fitzhugh–Nagumo equations have been analyzed. The novel algorithm has precision and stability comparable to those of pseudo-spectral codes, but is more convenient to be employed for systems with large linear extention L. As for finite-difference methods, the implementation of the present scheme requires only information about the local enviroment and this allows us to treat systems with very complicated boundary conditions.

In this paper, the novel exact solitary wave solutions for the generalized nonlinear Schrödinger equation with parabolic nonlinear (NL) law employing the improved $cosh(\mathrm{\Gamma }(\varpi ))-\mathrm{\text{sech}}(\mathrm{\Gamma }(\varpi ))$ function scheme and the combined $cos(\mathrm{\Gamma }(\varpi ))-sec(\mathrm{\Gamma }(\varpi ))$ function scheme are found. Diverse collections of hyperbolic and trigonometric function solutions acquired rely on a map between the considered equation and an auxiliary ODE. Received solutions are recast in several hyperbolic, rational and trigonometric forms based on different restrictions between parameters involved in equations and integration constants that appear in the solution. A few significant ones among the reported solutions are pictured to perceive the physical utility and peculiarity of the considered model utilizing mathematical software. The main subject of this work is that one can visualize and update the knowledge to overcome the most common techniques and defeat to solve the ODEs and PDEs. We demonstrated that these solutions validated the program using Maple and found them correct. The proposed methodology for solving the metamaterials model has been designed to be effectual, unpretentious, expedient and manageable. Applications of the solutions by the mentioned techniques will be useful to investigate the signals properties of optical fibers, plasma physics phenomena, electromagnetic fields occurrences and various types of nonlinear metamaterials models.