American Institute of Mathematical Sciences

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:

show all references

References:
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