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 nonlinear Schrödinger equation $(-i{\partial }_t+{\partial }_{xx})u=\pm (|u{|}^{4}u)$, 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.

References added/updated

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