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April  2019, 39(4): 1745-1777. doi: 10.3934/dcds.2019076

Vortex structures for some geometric flows from pseudo-Euclidean spaces

Ruiqi Jiang 1, , Youde Wang 2, and  Jun Yang 3,,

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

Received  October 2017 Revised  September 2018 Published  January 2019

Fund Project: 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.

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.

Keywords: Wave map equations, Schrödinger flows, Landau-Lifshitz equations, Vortices.
Mathematics Subject Classification: Primary: 35Q41, 58E50.
Citation: Ruiqi Jiang, Youde Wang, Jun Yang. Vortex structures for some geometric flows from pseudo-Euclidean spaces. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1745-1777. doi: 10.3934/dcds.2019076
References:
[1]

M. del Pino, P. 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 Pino, M. 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. Kenig, G. 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. Lin, W. 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.

show all references

References:
[1]

M. del Pino, P. 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 Pino, M. 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. Kenig, G. 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. Lin, W. 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.

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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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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