Rigorous justification of the short-pulse equation

TL;DR

This paper rigorously proves that the short-pulse equation, derived from Maxwell equations, accurately models small-norm solutions over finite times, validating its use in describing pulse dynamics.

Contribution

It provides a rigorous mathematical justification for the short-pulse equation from Maxwell equations, focusing on small-norm solutions and their validity over finite time intervals.

Findings

Small-norm solutions exist for infinite times including pulse interactions

Error bounds are controlled only over finite time intervals

The justification applies to solutions derived from Maxwell equations

Abstract

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.