On decay of shock-like waves into compact oscillons

TL;DR

This paper investigates how shock-like waves in the signum-Gordon model decay into oscillon-like structures, revealing a cascade process similar to oscillon scattering, with implications for understanding nonlinear wave dynamics.

Contribution

It introduces regularized initial data leading to finite-energy shock-like waves that decay into oscillons, a novel insight into wave decay mechanisms in the signum-Gordon model.

Findings

Shock-like waves exist temporarily and decay into oscillons.

Decay pattern resembles oscillon scattering processes.

Finite gradient energy waves are produced from regularized initial data.

Abstract

The signum-Gordon model in 1+1 dimensions possesses the exact shockwave solution with discontinuity of the field at the light cone and infinite gradient energy. The energy of a regular part of the wave inside the light cone is finite and it grows linearly with time. The initial data for such waves contain a field configuration which is null in the space and has time derivative proportional to the Dirac delta. We study regularized initial data that lead to shock-like waves with finite gradient energy. We found that such waves exist in the finite time intervals and finally they decay and produce a cascade of oscillon-like structures. A pattern of the decay is very similar to the one observed in process of scattering of compact oscillons.