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Vortex structures for some geometric flows from pseudo-Euclidean spaces

  • * Corresponding author: Jun Yang

    * Corresponding author: Jun Yang

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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  • 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.

    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}} $
    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)}} $
    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}} $
    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)}} $
    Type D $ \frac{c_4}{2\omega_6} $ $ \frac{2\omega_6}{\sqrt{1+|\omega_5|^2-|\omega_6|^2}} $
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