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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 |
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.
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. |
Equations | Sol. Type | ||
(11) | Type A | null | |
Type B | null | ||
(12) | Type A | null | |
Type B | null | ||
(13) | Type C | ||
Type D | |||
(14) | Type C | ||
Type D |
Equations | Sol. Type | ||
(11) | Type A | null | |
Type B | null | ||
(12) | Type A | null | |
Type B | null | ||
(13) | Type C | ||
Type D | |||
(14) | Type C | ||
Type D |
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