Vortex structures for some geometric flows from pseudo-Euclidean spaces

Ruiqi Jiang; Youde Wang; Jun Yang

doi:10.3934/dcds.2019076

Article Contents

2019, Volume 39, Issue 4: 1745-1777. Doi: 10.3934/dcds.2019076

This issue Previous Article Entropy rigidity and Hilbert volume Next Article Spectra of expanding maps on Besov spaces

Vortex structures for some geometric flows from pseudo-Euclidean spaces

1.
College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
2.
College of Mathematics and Information Sciences, Guangzhou University, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100190, China
3.
School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan 430079, China

^* Corresponding author: Jun Yang
^* Corresponding author: Jun Yang

Received: September 30, 2017

Revised: August 31, 2018

Published: January 2019

The first author is supported by the Fundamental Research Funds for the Central Universities as well as NSFC grant 11471316; The second author is supported by NSFC grant 11471316 and 11731001; The third author is supported by NSFC grant 11371254 and 11671144

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Abstract

For some geometric flows (such as wave map equations, Schrödinger flows) from pseudo-Euclidean spaces to a unit sphere contained in a three-dimensional Euclidean space, we construct solutions with various vortex structures (vortex pairs, elliptic/hyperbolic vortex circles, and also elliptic vortex helices). The approaches base on the transformations associated with the symmetries of the nonlinear problems, which will lead to two-dimensional elliptic problems with resolution theory given by the finite-dimensional Lyapunov-Schmidt reduction method in nonlinear analysis.

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Mathematics Subject Classification: Primary: 35Q41, 58E50.

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Table 1. The Expressions of Parameters $ \kappa $ and $ \mu $

Equations	Sol. Type	$ \kappa $	$ \mu $
(11)	Type A	null	$ \frac{2c_1}{\sqrt{1-\|c_1\|^2}} $
(11)	Type B	null	$ \frac{2\omega_3}{\sqrt{\|\omega_2\|^2-\|\omega_3\|^2}} $
(12)	Type A	null	$ \frac{2c_1\omega_1}{\sqrt{(1-\|c_1\|^2)(1+\|\omega_1\|^2)}} $
(12)	Type B	null	$ \frac{2\omega_3}{\sqrt{1+\|\omega_2\|^2-\|\omega_3\|^2}} $
(13)	Type C	$ \frac{c_3}{2c_2\omega_4} $	$ \frac{2c_2}{\sqrt{1-\|c_2\|^2}} $
(13)	Type D	$ \frac{c_4}{2\omega_6} $	$ \frac{2\omega_6}{\sqrt{\|\omega_5\|^2-\|\omega_6\|^2}} $
(14)	Type C	$ \frac{c_3}{2c_2\omega_4} $	$ \frac{2c_2\omega_4}{\sqrt{(1-\|c_2\|^2)(1+\|\omega_4\|^2)}} $
(14)	Type D	$ \frac{c_4}{2\omega_6} $	$ \frac{2\omega_6}{\sqrt{1+\|\omega_5\|^2-\|\omega_6\|^2}} $

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