Numerical adiabatic perturbation theory for the absolute |K|(p,p) equation

Abstract

In physical applications, the absolute |K|(p,p) equation should be preferred to the widely used Rosenau–Hyman K(p,p) equation due to the robustness of its compactons and anticompactons interactions observed in numerical simulations with small hyperviscosity. In order to understand the effect of the hyperviscosity in solutions with multiple compactons of the |K|(p,p) equation, the adiabatic perturbation theory has been applied. For a single compacton, this theory can be solved analytically showing that the second invariant decreases for p smaller than a critical value, as expected for a dissipative perturbation, but increases otherwise. This analytical prediction is in good agreement with the numerical results. In order to predict the evolution of the second invariant in time as a function of the hyperviscosity parameter for general solutions of the K(p,p) equation, a numerical implementation of the adiabatic perturbation theory has been developed. This adiabatic numerical prediction agrees with the evolution of the second invariant in the propagation of a single compacton, the generation of compacton trains from a truncated cosine initial condition, and compacton–compacton chase collisions. However, discrepancies emerge in other scenarios, such as the generation of a compacton train from a dilated compacton and in compacton–anticompacton chase collisions. Our findings support the use of the numerical adiabatic perturbation theory for analyzing the evolution of invariants due to hyperviscosity in multi-compacton simulations.