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

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