Note on scale invariance and self-similar evolution in (3+1)-dimensional signum-Gordon model

TL;DR

This paper explores self-similar solutions in a (3+1)-dimensional signum-Gordon model, revealing phenomena like field wipeout and energy accumulation, highlighting scale invariance and self-similar evolution in nonlinear wave equations.

Contribution

It introduces new classes of self-similar solutions in the signum-Gordon model, constructed from cubic polynomials in the scale-invariant variable, demonstrating diverse dynamical behaviors.

Findings

Solutions describe field wipeout and energy accumulation.

Solutions are constructed from cubic polynomials in t/r.

Highlights scale invariance in nonlinear wave evolution.

Abstract

Several classes of self-similar, spherically symmetric solutions of relativistic wave equation with nonlinear term of the form sign(\phi) are presented. They are constructed from cubic polynomials in the scale invariant variable t/r. One class of solutions describes a process of wiping out the initial field, another an accumulation of field energy in a finite and growing region of space.