On the mass-critical generalized KdV equation

TL;DR

This paper investigates the mass-critical generalized KdV equation, establishing a conditional framework for minimal-mass blowup solutions based on the validity of related conjectures, and classifies potential blowup solutions.

Contribution

It links the blowup behavior of the generalized KdV equation to the well-known NLS conjecture, providing a classification of minimal-mass blowup solutions under certain conditions.

Findings

Conditional existence of minimal-mass blowup solutions

Classification into self-similar, soliton-like, or cascade solutions

Dependence on the NLS well-posedness conjecture

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 nonlinear Schr\"odinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low frequency cascade solution.