# Ill-Posedness of the Third Order NLS Equation with Raman Scattering Term

## TL;DR

This paper investigates the mathematical properties of a third order nonlinear Schrödinger equation with Raman scattering, demonstrating ill-posedness in Sobolev spaces and establishing local well-posedness in analytic spaces.

## Contribution

It provides new insights into the well-posedness and ill-posedness of the equation, including nonexistence results and conditions for local existence.

## Key findings

- Nonexistence of solutions in Sobolev spaces
- Norm inflation of the data-solution map
- Local unique existence in analytic function space

## Abstract

We consider the ill-posedness and well-posedness of the Cauchy problem for the third order NLS equation with Raman scattering term on the one dimensional torus. It is regarded as a mathematical model for the photonic crystal fiber oscillator. Regarding the ill-posedness, we show the nonexistence of solutions in the Sobolev space and the norm inflation of the data-solution map under slightly different conditions, respectively. We also prove the local unique existence of solutions in the analytic function space.