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Description

We consider the nonlinear Schrödinger equation in $n$ space dimensions \[ iu_t + \Delta u + |u|^{p-1}u = 0, \;x \in \mathbb{R}^n,\; t > 0 \] and study the existence and stability of standing wave solutions of the form \[ \begin{cases} e^{iwt}e^{i \sum_{j=1}^k m_j \theta_j}\phi_w(r_1, r_2, \dots, r_k),& n = 2k\\ e^{iwt}e^{i \sum^k_{j=1} m_j \theta_j}\phi_w(r_1, r_2, \dots, r_k, z),& n = 2k + 1 \end{cases} \] for $n = 2k$, $(r_j,\theta_j)$ are polar coordinates in $\mathbb{R}^2$, $j = 1, 2,\dots, k$; for $n = 2k + 1$, $(r_j,\theta_j)$ are polar coordinates in $\mathbb{R}^2$, $(r_k,\theta_k,z)$ are cylindrical coordinates in $\mathbb{R}^3$, $j = 1, 2,\dots, k-1$. We show the existence of such solutions as minimizers of a constrained functional and conclude from there that such standing waves are stable if $1 < p < 1 + 4/n$.