CPAA
Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data
Hiroyuki Hirayama
Communications on Pure & Applied Analysis 2014, 13(4): 1563-1591 doi: 10.3934/cpaa.2014.13.1563
In the present paper, we consider the Cauchy problem of a system of quadratic derivative nonlinear Schrödinger equations which was introduced by M. Colin and T. Colin (2004) as a model of laser-plasma interaction. The local existence of the solution of the system in the Sobolev space $H^s$ for $s > d/2+3$ is proved by M. Colin and T. Colin. We prove the well-posedness of the system with low regularity initial data. For some cases, we also prove the well-posedness and the scattering at the scaling critical regularity by using $U^2$ space and $V^2$ space which are applied to prove the well-posedness and the scattering for KP-II equation at the scaling critical regularity by Hadac, Herr and Koch (2009).
keywords: Schr\"odinger equation scaling critical well-posedness Bilinear estimate Cauchy problem bounded $p$-variation.
CPAA
Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity
Hiroyuki Hirayama Mamoru Okamoto
Communications on Pure & Applied Analysis 2016, 15(3): 831-851 doi: 10.3934/cpaa.2016.15.831
In the present paper, we consider the Cauchy problem of fourth order nonlinear Schrödinger type equations with derivative nonlinearity. In one dimensional case, the small data global well-posedness and scattering for the fourth order nonlinear Schrödinger equation with the nonlinear term $\partial _x (\overline{u}^4)$ are shown in the scaling invariant space $\dot{H}^{-1/2}$. Furthermore, we show that the same result holds for the $d \ge 2$ and derivative polynomial type nonlinearity, for example $|\nabla | (u^m)$ with $(m-1)d \ge 4$, where $d$ denotes the space dimension.
keywords: scaling critical Cauchy problem bounded $p$-variation. Fourth order Schrödinger equation well-posedness
DCDS
Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity
Hiroyuki Hirayama Mamoru Okamoto
Discrete & Continuous Dynamical Systems - A 2016, 36(12): 6943-6974 doi: 10.3934/dcds.2016102
We consider the Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity $(i\partial _t + \Delta ) u= \pm \partial (\overline{u}^m)$ on $\mathbb{R} ^d$, $d \ge 1$, with random initial data, where $\partial$ is a first order derivative with respect to the spatial variable, for example a linear combination of $\frac{\partial}{\partial x_1} , \, \dots , \, \frac{\partial}{\partial x_d}$ or $|\nabla |= \mathcal{F}^{-1}[|\xi | \mathcal{F}]$. We prove that almost sure local in time well-posedness, small data global in time well-posedness and scattering hold in $H^s(\mathbb{R} ^d)$ with $s> \max \left( \frac{d-1}{d} s_c , \frac{s_c}{2}, s_c - \frac{d}{2(d+1)} \right)$ for $d+m \ge 5$, where $s$ is below the scaling critical regularity $s_c := \frac{d}{2}-\frac{1}{m-1}$.
keywords: Random data Cauchy problem Schrödinger equation.

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