Numerical interactions between compactons and kovatons of the Rosenau-Pikovsky K(cos) equation

A numerical study of the nonlinear wave solutions of the Rosenau-Pikovsky K(cos) equation is presented. This equation supports at least two kind of solitary waves with compact support: compactons of varying amplitude and speed, both bounded, and kovatons which have the maximum compacton amplitude, but arbitrary width. A new Pad\'e numerical method is used to simulate the propagation and, with small artificial viscosity added, the interaction between these kind of solitary waves. Several numerically induced phenomena that appear while propagating these compact travelling waves are discussed quantitatively, including self-similar forward and backward wavepackets. The collisions of compactons and kovatons show new phenomena such as the inversion of compactons and the generation of pairwise ripples decomposing into small compacton-anticompacton pairs

Dissipative perturbations for the K(n,n) Rosenau-Hyman equation

Compactons are compactly supported solitary waves for nondissipative evolution equations with nonlinear dispersion. In applications, these model equations are accompanied by dissipative terms which can be treated as small perturbations. We apply the method of adiabatic perturbations to compactons governed by the K(n,n) Rosenau-Hyman equation in the presence of dissipative terms preserving the "mass" of the compactons. The evolution equations for both the velocity and the amplitude of the compactons are determined for some linear and nonlinear dissipative terms: second-, fourth-, and sixth-order in the former case, and second- and fourth-order in the latter one. The numerical validation of the method is presented for a fourth-order, linear, dissipative perturbation which corresponds to a singular perturbation term

Self-similar Radiation from Numerical Rosenau-Hyman Compactons

The numerical simulation of compactons, solitary waves with compact support, is characterized by the presence of spurious phenomena, as numerically-induced radiation, which is illustrated here using four numerical methods applied to the Rosenau-Hyman K(p,p) equation. Both forward and backward radiations are emitted from the compacton presenting a self-similar shape which has been illustrated graphically by the proper scaling. A grid refinement study shows that the amplitude of the radiations decreases as the grid size does, confirming its numerical origin. The front velocity and the amplitude of both radiations have been studied as a function of both the compacton and the numerical parameters. The amplitude of the radiations decreases exponentially in time, being characterized by a nearly constant scaling exponent. An ansatz for both the backward and forward radiations corresponding to a self-similar function characterized by the scaling exponent is suggested by the present numerical results.Comment: To be published in Journal of Computational Physic

Fractal structure of the soliton scattering for the graphene superlattice equation

The graphene superlattice equation, a modified sine-Gordon equation, governs the propagation of solitary electromagnetic waves in a graphene superlattice. This equation has kink solutions without explicit analytical expression, requiring the use of quadrature methods. The inelastic collision of kinks and antikinks with the same but opposite speed is studied numerically for the first time; after their interaction they escape to infinity when its speed is either larger than a critical value or it is inside a series of resonance windows; otherwise, they form a breather-like state that slowly decays by radiating energy. Here, the fractal structure of these resonance windows is characterized by using a multi-index notation and their main features are compared with the predictions of the resonant energy exchange theory showing good agreement. Our results can be interpreted as new evidence in favour of this theory.Comment: 27 pages, 10 figures, 3 table