# H\"ormander's method for the characteristic Cauchy problem and conformal scattering for a non linear wave equation

## TL;DR

This paper proves the existence of a conformal scattering operator for a nonlinear wave equation on a non-stationary background by employing Hörmander's method to transform the characteristic initial value problem into a standard Cauchy problem.

## Contribution

It applies Hörmander's method to establish conformal scattering for a cubic defocusing wave equation in a non-stationary setting, extending previous techniques.

## Key findings

- Existence of a conformal scattering operator for the nonlinear wave equation.
- Successful application of Hörmander's method to a non-stationary background.
- Transformation of the characteristic problem into a standard Cauchy problem.

## Abstract

The purpose of this note is to prove the existence of a conformal scattering operator for the cubic defocusing wave equation on a non-stationary background. The proof essentially relies on solving the characteristic initial value problem by the method developed by H\"ormander. This method consists in slowing down the propagation speed of the waves to transform a characteristic initial value problem into a standard Cauchy problem.