论文标题
在具有浓缩非线性的Schroedinger方程的站立波上
On the standing waves of the Schroedinger equation with concentrated nonlinearity
论文作者
论文摘要
我们在$ {\ mathbf r^n} $上研究了浓缩的NLS，具有功率非线性，由分数laplacian驱动，$（ - δ）^s，s> \ frac {n} {2} $。我们在最佳参数范围内明确构建孤立波，以便它们属于自然能量空间$ h^s $。接下来，我们提供了它们的光谱稳定性的完整分类。最后，我们表明波浪是非分类的，因此，每当它们在频谱上稳定时，它们都是轨道稳定的。 顺便说一句，我们的构造表明，集中NLS的孤子曲线实际上是Sobolev嵌入$ h^s（{\ Mathbf r^n}）\ hookrightArrow l^\ infty（{\ Mathbf r^n}）$的精确最小化，这提供了这些替代的指导和杰出的构造，这些均具有这些替代的驱动力和公正性。
We study the concentrated NLS on ${\mathbf R^n}$, with power non-linearities, driven by the fractional Laplacian, $(-Δ)^s, s>\frac{n}{2}$. We construct the solitary waves explicitly, in an optimal range of the parameters, so that they belong to the natural energy space $H^s$. Next, we provide a complete classification of their spectral stability. Finally, we show that the waves are non-degenerate and consequently orbitally stable, whenever they are spectrally stable. Incidentally, our construction shows that the soliton profiles for the concentrated NLS are in fact exact minimizers of the Sobolev embedding $H^s({\mathbf R^n})\hookrightarrow L^\infty({\mathbf R^n})$, which provides an alternative calculation and justification of the sharp constants in these inequalities.
