arXiv 2303.033122023-03-06math.APmath.CA

平面上各向异性非线性薛定谔模型的孤波

On the solitary waves for anisotropic nonlinear Schrödinger models on the plane

Tianxiang Gou, Hichem Hajaiej, Atanas G. Stefanov

摘要：

讨论了聚焦各向异性非线性薛定谔方程\begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*}对$0<s<1$和$p>2$的影响。这里方程是各向异性的，这意味着解沿$x$轴和$y$轴的色散是不同的。我们发现，而本地化的时间周期波，即$u=e^{-\mathrm{i} ωt} ϕ$的形式的解决方案，不存在于制度$p\geq p_s:=\frac{2(1+s)}{1-s}$，他们确实存在于补充制度$2<p<p_s$。实际上，我们是用变分的方法来构造它们的，我们建立了一些关键的性质。重要的是，我们完全表征其光谱稳定性。我们的考虑很容易扩展到更高维的情况。我们还证明了在自然弱非简并假设下这些波的唯一性。对于接近$1$的$s$，实际上去除了这个假设，这意味着在整个参数范围内波是唯一的。

英文摘要：

The focussing anisotropic nonlinear Schrödinger equation \begin{align*} \mathrm{i} u_t-\partial_{xx} u + (-\partial_{yy})^s u=|u|^{p-2}u \quad \mbox{in}\ \mathbb{R} \times \mathbb{R}^2 \end{align*} is considered for $0<s<1$ and $p>2$. Here the equation is of anisotropy, it means that dispersion of solutions along $x$-axis and $y$-axis is different. We show that while localized time-periodic waves, that are solutions in the form $u=e^{-\mathrm{i} ωt} ϕ$, do not exist in the regime $p\geq p_s:=\frac{2(1+s)}{1-s}$, they do exist in the complementary regime $2<p<p_s$. In fact, we construct them variationally and we establish a number of key properties. Importantly, we completely characterize their spectral stability properties. Our consideration are easily extendable to the higher dimensional situation. We also show uniqueness of these waves under a natural weak non-degeneracy assumption. This assumption is actually removed for $s$ close to $1$, implying uniqueness for the waves in the full range of parameters.

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摘要：

英文摘要：

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