Classification of traveling waves for a quadratic Szeg{\"o} equation

TL;DR

This paper classifies all traveling wave solutions of a quadratic Szeg{"o} equation, identifying two families of rational functions and analyzing their stability properties.

Contribution

It provides a complete classification of traveling waves for the quadratic Szeg{"o} equation, including stability analysis of the identified solutions.

Findings

Traveling waves are given by two families of rational functions.

One family is generated by a stable ground state.

The other family is orbitally unstable.

Abstract

We give a complete classification of the traveling waves of the following quadratic Szeg{\"o} equation : $i \partial\_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u\_0$, and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.