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

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