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

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

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.

Citation: Ruiqi Jiang, Youde Wang, Jun Yang. Vortex structures for some geometric flows from pseudo-Euclidean spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1745-1777. doi: 10.3934/dcds.2019076
References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Hubert and R. Schafer, Magnetic Domains: The Analysis of Magnetic Microstructures, Berlin, Heidelberg, Springer-Verlag, 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

A. Hubert and R. Schafer, Magnetic Domains: The Analysis of Magnetic Microstructures, Berlin, Heidelberg, Springer-Verlag, 1998. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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}} $
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}} $
[1]

Wei Deng, Baisheng Yan. On Landau-Lifshitz equations of no-exchange energy models in ferromagnetics. Evolution Equations & Control Theory, 2013, 2 (4) : 599-620. doi: 10.3934/eect.2013.2.599

[2]

Ze Li, Lifeng Zhao. Convergence to harmonic maps for the Landau-Lifshitz flows between two dimensional hyperbolic spaces. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 607-638. doi: 10.3934/dcds.2019025

[3]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867

[4]

Xueke Pu, Boling Guo, Jingjun Zhang. Global weak solutions to the 1-D fractional Landau-Lifshitz equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 199-207. doi: 10.3934/dcdsb.2010.14.199

[5]

Jian Zhai, Zhihui Cai. $\Gamma$-convergence with Dirichlet boundary condition and Landau-Lifshitz functional for thin film. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 1071-1085. doi: 10.3934/dcdsb.2009.11.1071

[6]

Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644

[7]

Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919

[8]

Rémi Carles, Christof Sparber. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 759-774. doi: 10.3934/dcdsb.2012.17.759

[9]

Jaeyoung Byeon, Ohsang Kwon, Yoshihito Oshita. Standing wave concentrating on compact manifolds for nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 825-842. doi: 10.3934/cpaa.2015.14.825

[10]

P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure & Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029

[11]

Guangwu Wang, Boling Guo. Global weak solution to the quantum Navier-Stokes-Landau-Lifshitz equations with density-dependent viscosity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 6141-6166. doi: 10.3934/dcdsb.2019133

[12]

Akihiro Shimomura. Modified wave operators for the coupled wave-Schrödinger equations in three space dimensions. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1571-1586. doi: 10.3934/dcds.2003.9.1571

[13]

Thomas Duyckaerts, Carlos E. Kenig, Frank Merle. Profiles for bounded solutions of dispersive equations, with applications to energy-critical wave and Schrödinger equations. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1275-1326. doi: 10.3934/cpaa.2015.14.1275

[14]

Peng Gao, Yong Li. Averaging principle for the Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2147-2168. doi: 10.3934/dcdsb.2017089

[15]

Elena Cordero, Fabio Nicola, Luigi Rodino. Schrödinger equations with rough Hamiltonians. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4805-4821. doi: 10.3934/dcds.2015.35.4805

[16]

Renata Bunoiu, Radu Precup, Csaba Varga. Multiple positive standing wave solutions for schrödinger equations with oscillating state-dependent potentials. Communications on Pure & Applied Analysis, 2017, 16 (3) : 953-972. doi: 10.3934/cpaa.2017046

[17]

Tarek Saanouni. Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (1) : 273-291. doi: 10.3934/cpaa.2014.13.273

[18]

Noboru Okazawa, Toshiyuki Suzuki, Tomomi Yokota. Energy methods for abstract nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2012, 1 (2) : 337-354. doi: 10.3934/eect.2012.1.337

[19]

Nakao Hayashi, Tohru Ozawa. Schrödinger equations with nonlinearity of integral type. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 475-484. doi: 10.3934/dcds.1995.1.475

[20]

Rémi Carles, Clotilde Fermanian-Kammerer, Norbert J. Mauser, Hans Peter Stimming. On the time evolution of Wigner measures for Schrödinger equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 559-585. doi: 10.3934/cpaa.2009.8.559

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (89)
  • HTML views (63)
  • Cited by (0)

Other articles
by authors

[Back to Top]