# Smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation

## TL;DR

This paper constructs smooth travelling-wave solutions for the inviscid surface quasi-geostrophic equation, using variational methods to solve a fractional nonlinear elliptic equation, analogous to vortex pairs in Euler flows.

## Contribution

It introduces a novel method for constructing smooth travelling-wave solutions to the SQG equation via variational techniques.

## Key findings

- Existence of smooth travelling-wave solutions to the SQG equation.
- Solutions resemble vortex anti-vortex pairs in Euler flows.
- Method relies on solving a fractional nonlinear elliptic equation.

## Abstract

We construct families of smooth travelling-wave solutions to the inviscid surface quasi-geostrophic equation (SQG). These solutions can be viewed as the equivalents for this equation of the vortex anti-vortex pairs in the context of the incompressible Euler equation. Our argument relies on the stream function formulation and eventually amounts to solving a fractional nonlinear elliptic equation by variational methods.