# Coherent $\pi$-pulse emitted by a dense relativistic cold electron beam

## TL;DR

This paper demonstrates that a three-wave interaction system in free-electron lasers can be described by the Sine-Gordon equation, and provides numerical solutions for the system's behavior in space and time.

## Contribution

It establishes a connection between free-electron laser equations and the Sine-Gordon equation, offering a new analytical framework.

## Key findings

- The three-wave interaction equations satisfy the Sine-Gordon equation.
- Numerical solutions for the full space-time behavior are obtained.
- The model enhances understanding of coherent $mbda$-pulse emission.

## Abstract

Starting from a three-wave interaction system of equations for free-electron lasers in the framework of a quantum fluid model, we show that these equations satisfy the Sine-Gordon equation. The full solution in space and in time of this set of equations are numerically obtained.