A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
A sufficient condition for global existence of solutions to a generalized derivative nonlinear Schrödinger equation
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DOI：
10.2140/apde.2017.10.1149
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发表时间：
2016-10
期刊：
影响因子：
2.2
通讯作者：
Noriyoshi Fukaya;M. Hayashi;Takahisa Inui
中科院分区：
文献类型：
--
作者：
Noriyoshi Fukaya;M. Hayashi;Takahisa Inui
我们通过变分参数给出了广义导数非线性 Schr\"{o}dinger 方程 (gDNLS) 解的全局存在性的充分条件。变分参数适用于三次导数非线性 Schr\"{o}dinger 方程 (DNLS)。对于(DNLS)，吴在论文“Global Well-pedness on the您的非线性导数Schr\”odinger方程”中证明了如果$\left\Vert u_0 \right\Vert_{L^2}^2<4\pi$$\left\Vert u_0 \right\Vert_{L^2}^2<4\pi$时，初始数据$u_0$的解是全局的。变分论证为我们提供了 (DNLS) 全局存在性的另一个证明。此外，通过变分论证，我们可以证明，如果初始数据 $u_0$ 满足 $\left\Vert u_0 \right\Vert_{L^2}^2=4\pi$ 并且动量 $P(u_0)$ 为负，则 (DNLS) 的解是全局的。
We give a sufficient condition for global existence of the solutions to a generalized derivative nonlinear Schr\"{o}dinger equation (gDNLS) by a variational argument. The variational argument is applicable to a cubic derivative nonlinear Schr\"{o}dinger equation (DNLS). For (DNLS), Wu proved that the solution with the initial data $u_0$ is global if $\left\Vert u_0 \right\Vert_{L^2}^2<4\pi$ by the sharp Gagliardo--Nirenberg inequality in the paper "Global well-posedness on the derivative nonlinear Schr\"odinger equation", Analysis & PDE 8 (2015), no. 5, 1101--1112. The variational argument gives us another proof of the global existence for (DNLS). Moreover, by the variational argument, we can show that the solution to (DNLS) is global if the initial data $u_0$ satisfies that $\left\Vert u_0 \right\Vert_{L^2}^2=4\pi$ and the momentum $P(u_0)$ is negative.