On the mass-critical generalized KdV equation

Abstract

We consider the mass-critical generalized Korteweg--de Vries equation$$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$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 $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, 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.