The Initial-Value Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions
The Initial-Value Problem for the Cubic-Quintic NLS with Nonvanishing Boundary Conditions
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具有非零边界条件的三次五次NLS初值问题
DOI：
10.1137/17m1116702
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发表时间：
2018
影响因子：
2
通讯作者：
M. Vişan
中科院分区：
文献类型：
--
作者：
R. Killip;Jason Murphy;M. Vişan
本文考虑三维空间中一类五阶非线性Schrodinger方程$（i\partial_t+\Delta）\psi=\alpha_1\psi-\alpha_{3}\vert\psi\vert^2\psi+\alpha_5\vert\psi\vert^4\psi $的初值问题，其中|\psi（x）|\to c >0$ as $|X|\to\infty$.这里的$\alpha_1 $，$\alpha_3 $，$\alpha_5 $和$c$使得$\psi（x）\equiv c$是这个方程的能量稳定平衡解。将边界条件归一化为$\psi（x）\到1$，|X|在加权Sobolev空间中，我们研究了u= psi-1的初值问题，证明了小初值下的一个散射结果.
We consider the initial-value problem for the cubic-quintic nonlinear Schrodinger equation $ (i\partial_t+\Delta)\psi=\alpha_1 \psi-\alpha_{3}\vert \psi\vert^2 \psi+\alpha_5\vert \psi\vert^4 \psi $ in three spatial dimensions in the class of solutions with $|\psi(x)|\to c >0$ as $|x|\to\infty$. Here $\alpha_1$, $\alpha_3$, $\alpha_5$, and $c$ are such that $\psi(x)\equiv c$ is an energetically stable equilibrium solution to this equation. Normalizing the boundary condition to $\psi(x)\to 1$ as $|x|\to\infty$, we study the associated initial-value problem for $u=\psi-1$ and prove a scattering result for small initial data in a weighted Sobolev space.