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Instability of vortex solitons for 2D focusing NLS. (English) Zbl 1158.35089

The author studies a nonlinear Schrödinger equation, which after separation of variables becomes \[ \phi{''}+\frac{1}{r}\phi{'}-(\omega+\frac{m^2}{r^2})\phi +f(\phi)=0,\;r>0.\tag{a} \] Here \(r\) and \(\theta\) denote polar coordinates. First, a solution which becomes close to a related formula as \(m\to\infty\) is investigated. Then a related linearized operator is studied. This makes it possible to prove theorem 1, which is concerned with estimates of the solutions to (ref {a}). Finally, theorem 2, which states that there exists an \(m_{\star}\in\mathbb{N}\) such that, if \(m\geq{m_{\star}},\) a standing wave solution \(e^{i(\omega{t}+m\theta)}\phi_{\omega}\) is linearly unstable, is proved.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35B35 Stability in context of PDEs
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