Signum-Gordon shock waves in (2+1) and (3+1) dimensions

TL;DR

This paper presents new exact solutions for shock waves in the Signum-Gordon scalar field equation in higher dimensions, analyzing their energy, behavior when driven forces are removed, and resulting structures in simulations.

Contribution

It extends previous one-dimensional solutions to higher dimensions and explores the shock wave dynamics and structures in (2+1) and (3+1) dimensions.

Findings

Exact solutions for $n=2$ and $3$ dimensions.

Energy of shock waves scales with $n$-dimensional volume.

Disabling the driving delta function prevents wave propagation.

Abstract

This study introduces novel, exact solutions to the scalar field Signum-Gordon equation that feature a discontinuity near the light cone. These solutions, applicable in higher spatial dimensions ($n > 1$), extend previous limitations to one dimension. Our chosen ansatz leads to an ordinary equation with exact solutions obtained for $n = 2$ and $3$ spatial dimensions. The shock wave's energy trapped within the light cone is proportional to the wave's $n$-dimensional volume and the field discontinuity at the wavefront. The investigation delves further into the behavior of shock waves when their driving force, represented by a delta function at the light cone, is disabled. Disabling this delta function disrupts energy transfer, preventing the wave's propagation as predicted by analytical calculations. We identify the region within the light cones where the field remains unaffected. Two-dimensional ($n = 2$) simulations reveal the formation of intriguing structures upon source removal. These structures include a central, stable feature resembling an oscillon and a surrounding ring that breaks down into smaller oscillations.