Traveling vortex helices for Schrödinger map equations

Wei, Juncheng; Yang, Jun

doi:10.1090/tran/6379

Traveling vortex helices for Schrödinger map equations
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by Juncheng Wei and Jun Yang PDF

Trans. Amer. Math. Soc. 368 (2016), 2589-2622 Request permission

Abstract:

We construct traveling wave solutions with vortex helix structures for the Schrödinger map equation \[ \frac {\partial m}{\partial t}= m \times (\Delta m - m_3 \vec {e}_3) \quad \mbox {on} \ {\mathbb R}^3 \times {\mathbb R} \] of the form $m(s_1,s_2, s_3 -\delta |\log \epsilon |\epsilon t)$ with traveling velocity $\delta |\log \epsilon |\epsilon$ along the direction of the $s_3$ axis. We use a perturbation approach which gives a complete characterization of the asymptotic behavior of the solutions.

References

S. V. Alekseenko, P. A. Kuibin, and V. L. Okulov, Theory of concentrated vortices, Springer, Berlin, 2007. An introduction; Translated from the 2003 Russian original. MR 2398034
P.-L. Sulem, C. Sulem, and C. Bardos, On the continuous limit for a system of classical spins, Comm. Math. Phys. 107 (1986), no. 3, 431–454. MR 866199
Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations 1 (1993), no. 2, 123–148. MR 1261720, DOI 10.1007/BF01191614
Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 13, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1269538, DOI 10.1007/978-1-4612-0287-5
Fabrice Béthuel, Philippe Gravejat, and Jean-Claude Saut, Existence and properties of travelling waves for the Gross-Pitaevskii equation, Stationary and time dependent Gross-Pitaevskii equations, Contemp. Math., vol. 473, Amer. Math. Soc., Providence, RI, 2008, pp. 55–103. MR 2522014, DOI 10.1090/conm/473/09224
Fabrice Béthuel, Philippe Gravejat, and Jean-Claude Saut, Travelling waves for the Gross-Pitaevskii equation. II, Comm. Math. Phys. 285 (2009), no. 2, 567–651. MR 2461988, DOI 10.1007/s00220-008-0614-2
F. Bethuel, G. Orlandi, and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 1, 17–94. MR 2041006
Fabrice Bethuel and Jean-Claude Saut, Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), no. 2, 147–238 (English, with English and French summaries). MR 1669387
Nai-Heng Chang, Jalal Shatah, and Karen Uhlenbeck, Schrödinger maps, Comm. Pure Appl. Math. 53 (2000), no. 5, 590–602. MR 1737504, DOI 10.1002/(SICI)1097-0312(200005)53:5<590::AID-CPA2>3.3.CO;2-I
David Chiron, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two, Nonlinear Anal. 58 (2004), no. 1-2, 175–204. MR 2070812, DOI 10.1016/j.na.2003.10.028
David Chiron, Vortex helices for the Gross-Pitaevskii equation, J. Math. Pures Appl. (9) 84 (2005), no. 11, 1555–1647 (English, with English and French summaries). MR 2181460, DOI 10.1016/j.matpur.2005.08.008
Jaigyoung Choe and Jens Hoppe, Higher dimensional minimal submanifolds generalizing the catenoid and helicoid, Tohoku Math. J. (2) 65 (2013), no. 1, 43–55. MR 3049639, DOI 10.2748/tmj/1365452624
Manuel del Pino, MichałKowalczyk, and Monica Musso, Variational reduction for Ginzburg-Landau vortices, J. Funct. Anal. 239 (2006), no. 2, 497–541. MR 2261336, DOI 10.1016/j.jfa.2006.07.006
Manuel del Pino, Monica Musso, and Frank Pacard, Solutions of the Allen-Cahn equation which are invariant under screw-motion, Manuscripta Math. 138 (2012), no. 3-4, 273–286. MR 2916313, DOI 10.1007/s00229-011-0492-3
Boling Guo and Shijin Ding, Initial-boundary value problem for the Landau-Lifshitz system. I. Existence and partial regularity, Progr. Natur. Sci. (English Ed.) 8 (1998), no. 1, 11–23. MR 1612862
Boling Guo and Shijin Ding, Initial-boundary value problem for the Landau-Lifshitz system. II. Uniqueness, Progr. Natur. Sci. (English Ed.) 8 (1998), no. 2, 147–151. MR 1656456
Weiyue Ding, On the Schrödinger flows, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 283–291. MR 1957040
Weiyue Ding, Hongyan Tang, and Chongchun Zeng, Self-similar solutions of Schrödinger flows, Calc. Var. Partial Differential Equations 34 (2009), no. 2, 267–277. MR 2448652, DOI 10.1007/s00526-008-0198-x
Weiyue Ding and Youde Wang, Schrödinger flow of maps into symplectic manifolds, Sci. China Ser. A 41 (1998), no. 7, 746–755. MR 1633799, DOI 10.1007/BF02901957
Weiyue Ding and Youde Wang, Local Schrödinger flow into Kähler manifolds, Sci. China Ser. A 44 (2001), no. 11, 1446–1464. MR 1877231, DOI 10.1007/BF02877074
Bo Ling Guo and Min Chun Hong, The Landau-Lifshitz equation of the ferromagnetic spin chain and harmonic maps, Calc. Var. Partial Differential Equations 1 (1993), no. 3, 311–334. MR 1261548, DOI 10.1007/BF01191298
Stephen Gustafson, Kyungkeun Kang, and Tai-Peng Tsai, Asymptotic stability of harmonic maps under the Schrödinger flow, Duke Math. J. 145 (2008), no. 3, 537–583. MR 2462113, DOI 10.1215/00127094-2008-058
S. Gustafson, K. Kang, and T.-P. Tsai, Schrödinger flow near harmonic maps, Comm. Pure Appl. Math. 60 (2007), no. 4, 463–499. MR 2290708, DOI 10.1002/cpa.20143
Stephen Gustafson and Jalal Shatah, The stability of localized solutions of Landau-Lifshitz equations, Comm. Pure Appl. Math. 55 (2002), no. 9, 1136–1159. MR 1908745, DOI 10.1002/cpa.3024
Feng Bo Hang and Fang Hua Lin, Static theory for planar ferromagnets and antiferromagnets, Acta Math. Sin. (Engl. Ser.) 17 (2001), no. 4, 541–580. MR 1891748, DOI 10.1007/s101140100136
Fengbo Hang and Fanghua Lin, A Liouville type theorem for minimizing maps, Methods Appl. Anal. 9 (2002), no. 3, 407–424. Special issue dedicated to Daniel W. Stroock and Srinivasa S. R. Varadhan on the occasion of their 60th birthday. MR 2023133, DOI 10.4310/MAA.2002.v9.n3.a7
A. Hubert and R. Schafer, Magnetic Domain—The Analysis of Magnetic Microstructures, Springer-Verlag, Berlin-Heidelberg, 1998.
J. Huhtamaki and P. Kuopanportti, Helical spin textures in dipolar Bose-Einstein condensates, Physical Review A 82 (2010), 053616.
C. A. Jones and P. H. Roberts, Motion in a Bose condensate IV, Axisymmetric solitary waves, J. Phys. A 15 (1982), 2599-2619.
S. Komineas and N. Papanicolaou, Topology and dynamics in ferromagnetic media, Phys. D 99 (1996), no. 1, 81–107. MR 1420808, DOI 10.1016/S0167-2789(96)00130-3
S. Komineas and N. Papanicolaou, Vortex dynamics in two-dimensional antiferromagnets, Nonlinearity 11 (1998), no. 2, 265–290. MR 1610760, DOI 10.1088/0951-7715/11/2/005
L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sow. 8 (1935), 153-169.
F. H. Lin and J. Shatah, Soliton dynamics in planar ferromagnets and anti-ferromagnets, Journal of Zhejiang University Science 4 (2003), no. 5, 503-510.
Fanghua Lin and Juncheng Wei, Traveling wave solutions of the Schrödinger map equation, Comm. Pure Appl. Math. 63 (2010), no. 12, 1585–1621. MR 2742008, DOI 10.1002/cpa.20338
F.-H. Lin and J. X. Xin, On the dynamical law of the Ginzburg-Landau vortices on the plane, Comm. Pure Appl. Math. 52 (1999), no. 10, 1189–1212. MR 1699965, DOI 10.1002/(SICI)1097-0312(199910)52:10<1189::AID-CPA1>3.0.CO;2-T
R. Moser, Partial regularity for Landau-Lifshitz equation, Preprint Series, Max-Planck-Institute for Mathematics in the Sciences 26/2002.
John C. Neu, Vortices in complex scalar fields, Phys. D 43 (1990), no. 2-3, 385–406. MR 1067918, DOI 10.1016/0167-2789(90)90143-D
N. Papanicolaou and P. N. Spathis, Semitopological solitons in planar ferromagnets, Nonlinearity 12 (1999), no. 2, 285–302. MR 1677795, DOI 10.1088/0951-7715/12/2/008
Jalal Shatah and Chongchun Zeng, Schrödinger maps and anti-ferromagnetic chains, Comm. Math. Phys. 262 (2006), no. 2, 299–315. MR 2200262, DOI 10.1007/s00220-005-1490-7
Changyou Wang, On Landau-Lifshitz equation in dimensions at most four, Indiana Univ. Math. J. 55 (2006), no. 5, 1615–1644. MR 2270931, DOI 10.1512/iumj.2006.55.2810
J. D. Watson and F. H. C. Crick, Molecular structure of nucleic acids: a structure for deoxyribose nucleic acid, Nature 171 (1953), 737-738.
A. Visintin, On Landau-Lifshitz’ equations for ferromagnetism, Japan J. Appl. Math. 2 (1985), no. 1, 69–84. MR 839320, DOI 10.1007/BF03167039
Yu Lin Zhou and Bo Ling Guo, Existence of weak solution for boundary problems of systems of ferro-magnetic chain, Sci. Sinica Ser. A 27 (1984), no. 8, 799–811. MR 795163
Yu Lin Zhou, Bo Ling Guo, and Shao Bin Tan, Existence and uniqueness of smooth solution for system of ferro-magnetic chain, Sci. China Ser. A 34 (1991), no. 3, 257–266. MR 1110342