On the mass-critical generalized KdV equation
Rowan Killip Soonsik Kwon Shuanglin Shao Monica Visan
Discrete & Continuous Dynamical Systems - A 2012, 32(1): 191-221 doi: 10.3934/dcds.2012.32.191
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: $L^2$-critical. Korteweg--de Vries equation
Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$
J. Colliander Justin Holmer Monica Visan Xiaoyi Zhang
Communications on Pure & Applied Analysis 2008, 7(3): 467-489 doi: 10.3934/cpaa.2008.7.467
We consider the Cauchy problem for a family of semilinear defocusing Schrödinger equations with monomial nonlinearities in one space dimension. We establish global well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz estimate giving a priori $L_{t,x}^8$ spacetime control on solutions.
keywords: scattering well-posedness Nonlinear Schrödinger equation Morawetz inequality.
The energy-critical NLS with inverse-square potential
Rowan Killip Changxing Miao Monica Visan Junyong Zhang Jiqiang Zheng
Discrete & Continuous Dynamical Systems - A 2017, 37(7): 3831-3866 doi: 10.3934/dcds.2017162

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

keywords: Nonlinear Schröodinger equation scattering inverse-square potential concentration compactness
The radial mass-subcritical NLS in negative order Sobolev spaces
Rowan Killip Satoshi Masaki Jason Murphy Monica Visan
Discrete & Continuous Dynamical Systems - A 2019, 39(1): 553-583 doi: 10.3934/dcds.2019023

We consider the mass-subcritical NLS in dimensions $d≥ 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.

keywords: Nonlinear Schrödinger equation scattering mass-subcritical

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