论文标题
thepole图上的Quintic NLS方程的站立波
Standing waves of the quintic NLS equation on the tadpole graph
论文作者
论文摘要
t图由一个圆和一个连接的顶点组成。我们分析了具有五重量功率非线性的非线性schrödinger方程的站立波，配备了Neumann-Kirchhoff边界条件。带有频率$ω\ in（ - \ infty，0）$的频率$ω\的剖面的曲线被特征在于$ l^6 $以限制为单位球的二次能量部分的全局最小化器。一组最小化器包括系统的基础状态，它们是恒定质量（$ l^2 $ norm）的能量的全球最小化器，但实际上更宽。尽管仅在一定的质量间隔内存在地面状态，但在（ - \ infty，0）$中的每个$ω\中都存在站立波，并且对应于较大的质量间隔。结果表明，存在关键频率$ω_0$和$ω_1$，因此，常驻波是$ω\ in [ω_0,0）$的基础状态，本地的能量最小化的能量以$ω\ in（ω_1，Ω__0）$的$ω\ in（ω_1，ω__0）$，以及convant constant oss $ω\ for $ω\ forcy for $ω\ fors（ - 证明利用了差分方程的变分方法和分析理论。
The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schrödinger equation with quintic power nonlinearity equipped with the Neumann-Kirchhoff boundary conditions at the vertex. The profile of the standing wave with the frequency $ω\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of minimizers includes the set of ground states of the system, which are the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the standing waves exist for every $ω\in (-\infty,0)$ and correspond to a bigger interval of masses. It is shown that there exist critical frequencies $ω_0$ and $ω_1$ such that the standing waves are the ground states for $ω\in [ω_0,0)$, local minimizers of the energy at constant mass for $ω\in (ω_1,ω_0)$, and saddle points of the energy at constant mass for $ω\in (-\infty,ω_1)$. Proofs make use of both the variational methods and the analytical theory for differential equations.
