Dynamical behavior near explicit self-similar blow up solutions for the Born-Infeld equation

TL;DR

This paper analyzes the stability of explicit self-similar blow-up solutions in the one-dimensional Born-Infeld equation, revealing their nonlinear stability within a specific region and connecting these solutions to models in electromagnetism and string theory.

Contribution

It introduces a new family of explicit self-similar solutions for the Born-Infeld equation and establishes their nonlinear stability in a certain domain.

Findings

Both the Born-Infeld equation and the linear wave equation admit the same self-similar blow-up solutions.

The self-similar blow-up solutions are nonlinearly stable inside a proper subset of the backward light cone.

The results connect the behavior of the Born-Infeld model to classical wave dynamics.

Abstract

This paper studies the dynamical behavior near a new family of explicit self-similar solutions for the one dimensional Born-Infeld equation. This quasilinear scalar field equation arises from nonlinear electromagnetism, as well as branes in string theory and minimal surfaces in Minkowski spacetimes. We show that both this model and the linear wave equation admit the same family of explicit timelike self-similar blow up solutions, meanwhile, Lyapunov nonlinear stability of those self-similar blow up solutions are given inside a strictly proper subset of the backward light cone.