# Hodograph solutions of the wave equation of nonlinear electrodynamics in the quantum vacuum

## TL;DR

This paper derives exact solutions for nonlinear electromagnetic wave interactions in vacuum, modeled by the Euler-Heisenberg Lagrangian, using hodograph transformations to linearize complex PDEs in a one-dimensional setting.

## Contribution

It introduces a novel application of hodograph solutions to the wave equation in nonlinear electrodynamics within quantum vacuum, providing explicit analytical solutions.

## Key findings

- Exact solutions for photon-photon scattering in vacuum
- Demonstration of hodograph transformation effectiveness
- Insights into nonlinear wave interactions in quantum electrodynamics

## Abstract

The process of photon-photon scattering in vacuum is investigated analytically in the long-wavelength limit within the framework of the Euler-Heisenberg Lagrangian. In order to solve the nonlinear partial differential equations (PDEs) obtained from this Lagrangian use is made of the hodograph transformation. This transformation makes it possible to turn a system of quasilinear PDEs into a system of linear PDEs. Exact solutions of the equations describing the nonlinear interaction of electromagnetic waves in vacuum in a one-dimensional configuration are obtained and analyzed.