On the mass-critical generalized KdV equation

Rowan Killip; Soonsik Kwon; Shuanglin Shao; Monica Visan

doi:10.3934/dcds.2012.32.191

Article Contents

2012, Volume 32, Issue 1: 191-221. Doi: 10.3934/dcds.2012.32.191

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On the mass-critical generalized KdV equation

1.
Mathematics Department, University of California, Los Angeles, Box 951555, Los Angeles, CA 90095
2.
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon, Korea 305-701
3.
Institute for Mathematics and its Applications, University of Minnesota, 207 Church St. SE Minneapolis, MN 55455

Received: August 2010

Revised: December 2010

Published: January 2012

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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 $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, 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.

Keywords:

Korteweg--de Vries equation,
$L^2$-critical.

Mathematics Subject Classification: Primary: 35Q53.

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