Field-theoretic Models with V-shaped Potentials

TL;DR

This paper explores field-theoretic models with V-shaped potentials, highlighting their unique static solutions, scaling invariance in small perturbations, and the existence of self-similar and shock wave solutions.

Contribution

It introduces the concept of V-shaped potentials in field theories and analyzes their static solutions, scaling properties, and dynamic phenomena such as shock waves.

Findings

Exact ground state achieved at finite distance with no exponential tails

Scaling invariance in small amplitude perturbations

Existence of self-similar and shock wave solutions

Abstract

In this lecture we outline the main results of our investigations of certain field-theoretic systems which have V-shaped field potential. After presenting physical examples of such systems, we show that in static problems the exact ground state value of the field is achieved on a finite distance - there are no exponential tails. This applies in particular to soliton-like object called the topological compacton. Next, we discuss scaling invariance which appears when the fields are restricted to small amplitude perturbations of the ground state. Evolution of such perturbations is governed by nonlinear equation with a non-smooth term which can not be linearized even in the limit of very small amplitudes. Finally, we briefly describe self-similar and shock wave solutions of that equation.