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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.

    Citation:

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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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  • [1] M. del PinoP. Felmer and M. Musso, Two-bubble solutions in the super-critical Bahri-Coron's problem, Calc. Var. Partial Differential Equations, 16 (2003), 113-145.  doi: 10.1007/s005260100142.
    [2] M. del PinoM. Kowalczyk and M. Musso, Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal., 239 (2006), 497-541.  doi: 10.1016/j.jfa.2006.07.006.
    [3] W. Y. Ding and Y. D. Wang, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A, 41 (1998), 746-755.  doi: 10.1007/BF02901957.
    [4] W. Y. Ding and H. Yin, Special periodic solutions of Schrödinger flow, Math. Z., 253 (2006), 555-570.  doi: 10.1007/s00209-005-0922-6.
    [5] C. Gui and J. Wei, Multiple interior peak solutions for some singularly perturbed Neumann problems, J. Differential Equations, 158 (1999), 1-27.  doi: 10.1016/S0022-0396(99)80016-3.
    [6] F. Hang and F. H. Lin, Static theory for planar ferromagnets an antiferromagnets, Acta Math. Sin. (Engl. Ser.), 17 (2001), 541-580.  doi: 10.1007/s101140100136.
    [7] F. Hang and F. H. Lin, A Liouville type theorem for minimizing maps, Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday, Methods Appl. Anal., 9 (2002), 407-424.  doi: 10.4310/MAA.2002.v9.n3.a7.
    [8] A. Hubert and R. Schafer, Magnetic Domains: The Analysis of Magnetic Microstructures, Berlin, Heidelberg, Springer-Verlag, 1998.
    [9] C. E. KenigG. Ponce and L. Vega, Smoothing effects and local existence theory for the generalized nonlinear Schrödinger equations, Invent. Math., 134 (1998), 489-545.  doi: 10.1007/s002220050272.
    [10] F. H. LinW. M. Ni and J. Wei, On the number of interior peak solutions for a singularly perturbed Neumann problem, Comm. Pure Appl. Math., 60 (2007), 252-281.  doi: 10.1002/cpa.20139.
    [11] F. H. Lin and J. Wei, Traveling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621.  doi: 10.1002/cpa.20338.
    [12] R. López, Constant mean curvature surfaces foliated by circles in Lorentz-Minkowski space, Geom. Dedicata, 76 (1999), 81-95.  doi: 10.1023/A:1005145820971.
    [13] N. Papanicolaou and P. N. Spathis, Semitopological solitons in planar ferromagnets, Nonlinearity, 12 (1999), 285-302.  doi: 10.1088/0951-7715/12/2/008.
    [14] C. Song and Y. D. Wang, Schrödinger soliton from Lorentzian manifolds, Acta Math. Sin. (Engl. Ser.), 27 (2011), 1455-1476.  doi: 10.1007/s10114-011-0229-y.
    [15] C. Song, X. Sun and Y. D. Wang, Geometric solitons of Hamiltonian flows on manifolds, J. Math. Phys., 54 (2013), 121505, 17pp. doi: 10.1063/1.4848775.
    [16] J. Wei and J. Yang, Traveling vortex helices for Schrödinger map equations, Trans. Amer. Math. Soc., 368 (2016), 2589-2622.  doi: 10.1090/tran/6379.
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