3 years ago

Orbital instability of standing waves for NLS equation on Star Graphs.

Adilbek Kairzhan

We consider a nonlinear Schr\"{o}dinger (NLS) equation with any positive power nonlinearity on a star graph $\Gamma$ ($N$ half-lines glued at the common vertex) with a $\delta$ interaction at the vertex. The strength of the interaction is defined by a fixed value $\alpha \in \mathbb{R}$. In the recent works of Adami {\it et al.}, it was shown that for $\alpha \neq 0$ the NLS equation on $\Gamma$ admits the unique symmetric (with respect to permutation of edges) standing wave and that all other possible standing waves are nonsymmetric. Also, it was proved for $\alpha<0$ that, in the NLS equation with a subcritical power-type nonlinearity, the unique symmetric standing wave is orbitally stable.

In this paper, we analyze stability of standing waves for both $\alpha<0$ and $\alpha>0$. By extending the Sturm theory to Schr\"{o}dinger operators on the star graph, we give the explicit count of the Morse and degeneracy indices for each standing wave. For $\alpha<0$, we prove that all nonsymmetric standing waves in the NLS equation with any positive power nonlinearity are orbitally unstable. For $\alpha>0$, we prove the orbital instability of all standing waves.

Publisher URL: http://arxiv.org/abs/1712.02773

DOI: arXiv:1712.02773v2

You might also like
Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

  • Download from Google Play
  • Download from App Store
  • Download from AppInChina

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.