On the mass-critical generalized KdV equation

We consider the mass-critical generalized Korteweg--de Vries equation(∂ₜ + ∂ₓₓₓ)u=± ∂ₓ(u⁵)for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed,then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinearSchrödinger equation (-i∂ₜ + ∂ₓₓ)u=± (|u|⁴u), there exists a minimal-mass blowup solution to themass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we canguarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-lowfrequency cascade solution.