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Stability of vortex solitons for $n$-dimensional focusing NLS Stanislavova, Milena
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 $1 < p < 1 + 4/n$.
Item Metadata
Title |
Stability of vortex solitons for $n$-dimensional focusing NLS
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Creator | |
Publisher |
Banff International Research Station for Mathematical Innovation and Discovery
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Date Issued |
2017-06-19T12:33
|
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 $1 < p < 1 + 4/n$.
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Extent |
22 minutes
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Subject | |
Type | |
File Format |
video/mp4
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Language |
eng
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Notes |
Author affiliation: University of Kansas
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Series | |
Date Available |
2017-12-16
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Provider |
Vancouver : University of British Columbia Library
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Rights |
Attribution-NonCommercial-NoDerivatives 4.0 International
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DOI |
10.14288/1.0362065
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URI | |
Affiliation | |
Peer Review Status |
Unreviewed
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Scholarly Level |
Faculty
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Rights URI | |
Aggregated Source Repository |
DSpace
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Item Media
Item Citations and Data
Rights
Attribution-NonCommercial-NoDerivatives 4.0 International