Scaling, self-similar solutions and shock waves for V-shaped field potentials

TL;DR

This paper explores a (1+1)-dimensional nonlinear field model with a V-shaped potential, revealing self-similar and shock wave solutions, and highlighting its scaling symmetry and universal properties.

Contribution

It introduces a universal V-shaped potential model with scaling symmetry and derives self-similar and shock wave solutions, extending understanding of nonlinear field theories.

Findings

Discovery of self-similar solutions in the model

Identification of shock wave solutions

Highlighting the model's scaling symmetry

Abstract

We investigate a (1+1)-dimensional nonlinear field theoretic model with the field potential $V(\phi)| = |\phi|.$ It can be obtained as the universal small amplitude limit in a class of models with potentials which are symmetrically V-shaped at their minima, or as a continuum limit of certain mechanical system with infinite number of degrees of freedom. The model has an interesting scaling symmetry of the 'on shell' type. We find self-similar as well as shock wave solutions of the field equation in that model.