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May  2014, 34(5): 2359-2388. doi: 10.3934/dcds.2014.34.2359

Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity

Jun Yang 1,

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

Received  February 2013 Revised  August 2013 Published  October 2013

By a perturbation approach, we construct traveling solitary solutions with various vortex structures(vortex pairs, vortex rings) for Klein-Gordon equation with Ginzburg-Landau nonlinearities.
Keywords: vortex rings, traveling solitary waves., Klein-Gordon equations, vortex pairs.
Mathematics Subject Classification: Primary: 35J25, 35J20; Secondary: 35B06, 35B4.
Citation: Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359
References:
[1]

F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), 123-148. doi: 10.1007/BF01191614.  Google Scholar

[2]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Inc., Boston, 1994. doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[3]

F. Bethuel, P. Gravejat and J. Saut, Existence and Properties of Travelling Waves for the Gross-Pitaevskii Equation, Stationary and time dependent Gross-Pitaevskii equations, 55-103, Contemp. Math., 473, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09224.  Google Scholar

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F. Bethuel, P. Gravejat and J.-G. Saut, Travelling waves for the Gross-Pitaevskii equation, II, Comm. Math. Phys., 285 (2009), 567-651. doi: 10.1007/s00220-008-0614-2.  Google Scholar

[5]

F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94.  Google Scholar

[6]

F. Bethuel and J. C. Saut, Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. H. Poincaré Phys. Theór., 70 (1999), 147-238.  Google Scholar

[7]

X. Chen, C. M. Elliott and Q. Tang, Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1075-1088. doi: 10.1017/S0308210500030122.  Google Scholar

[8]

D. Chiron, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two, Nonlinear Anal., 58 (2004), 175-204. doi: 10.1016/j.na.2003.10.028.  Google Scholar

[9]

D. Chiron, Vortex helices for the Gross-Pitaevskii equation, J. Math. Pures Appl. (9), 84 (2005), 1555-1647. doi: 10.1016/j.matpur.2005.08.008.  Google Scholar

[10]

M. del Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., 2004 (2004), 1511-1527. doi: 10.1155/S1073792804133588.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

F. Hang and F. H. Lin, Static theory for planar ferromagnets and antiferromagnets, Acta Math. Sin. (Engl. Ser.), 17 (2001), 541-580. doi: 10.1007/s101140100136.  Google Scholar

[15]

F. Hang and F. 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

[16]

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

[17]

C. A. Jones and P. H. Roberts, Motion in a Bose condensate IV, Axisymmetric solitary waves, J. Phys. A, 15 (1982), 2599-2619. doi: 10.1088/0305-4470/15/8/036.  Google Scholar

[18]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sow., 8 (1935), 153-169. Google Scholar

[19]

C. Lin and K. Wu, Singular limits of the Klein-Gordon equation, Arch. Rational Mech. Anal., 197 (2010), 689-711. doi: 10.1007/s00205-010-0324-8.  Google Scholar

[20]

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.  Google Scholar

[21]

F. H. Lin and J. Shatah, Soliton dynamics in planar ferromagnets and anti-ferromagnets, Journal of Zhejiang University Science, 4 (2003), 503-510. Google Scholar

[22]

F. Lin and J. Wei, Travelling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621. doi: 10.1002/cpa.20338.  Google Scholar

[23]

T.-C. Lin, J. Wei and J. Yang, Vortex rings for the Gross-Pitaevskii equation in $\mathbb R^3$, J. Math. Pures Appl., 100 (2013), 69-112. doi: 10.1016/j.matpur.2012.10.012.  Google Scholar

[24]

J. C. Neu, Vortices in complex scalar fields, Phy. D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[25]

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

[26]

Y. Yu, Vortex dynamics for nonlinear Klein-Gordon equation, J. Differential Equations, 251 (2011), 970-994. doi: 10.1016/j.jde.2011.04.023.  Google Scholar

show all references

References:
[1]

F. Bethuel, H. Brezis and F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional, Calc. Var. Partial Differential Equations, 1 (1993), 123-148. doi: 10.1007/BF01191614.  Google Scholar

[2]

F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser, Inc., Boston, 1994. doi: 10.1007/978-1-4612-0287-5.  Google Scholar

[3]

F. Bethuel, P. Gravejat and J. Saut, Existence and Properties of Travelling Waves for the Gross-Pitaevskii Equation, Stationary and time dependent Gross-Pitaevskii equations, 55-103, Contemp. Math., 473, Amer. Math. Soc., Providence, RI, 2008. doi: 10.1090/conm/473/09224.  Google Scholar

[4]

F. Bethuel, P. Gravejat and J.-G. Saut, Travelling waves for the Gross-Pitaevskii equation, II, Comm. Math. Phys., 285 (2009), 567-651. doi: 10.1007/s00220-008-0614-2.  Google Scholar

[5]

F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., 6 (2004), 17-94.  Google Scholar

[6]

F. Bethuel and J. C. Saut, Travelling waves for the Gross-Pitaevskii equation. I, Ann. Inst. H. Poincaré Phys. Theór., 70 (1999), 147-238.  Google Scholar

[7]

X. Chen, C. M. Elliott and Q. Tang, Shooting method for vortex solutions of a complex-valued Ginzburg-Landau equation, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 1075-1088. doi: 10.1017/S0308210500030122.  Google Scholar

[8]

D. Chiron, Travelling waves for the Gross-Pitaevskii equation in dimension larger than two, Nonlinear Anal., 58 (2004), 175-204. doi: 10.1016/j.na.2003.10.028.  Google Scholar

[9]

D. Chiron, Vortex helices for the Gross-Pitaevskii equation, J. Math. Pures Appl. (9), 84 (2005), 1555-1647. doi: 10.1016/j.matpur.2005.08.008.  Google Scholar

[10]

M. del Pino, P. Felmer and M. Kowalczyk, Minimality and nondegeneracy of degree-one Ginzburg-Landau vortex as a Hardy's type inequality, Int. Math. Res. Not., 2004 (2004), 1511-1527. doi: 10.1155/S1073792804133588.  Google Scholar

[11]

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.  Google Scholar

[12]

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.  Google Scholar

[13]

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

[14]

F. Hang and F. H. Lin, Static theory for planar ferromagnets and antiferromagnets, Acta Math. Sin. (Engl. Ser.), 17 (2001), 541-580. doi: 10.1007/s101140100136.  Google Scholar

[15]

F. Hang and F. 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

[16]

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

[17]

C. A. Jones and P. H. Roberts, Motion in a Bose condensate IV, Axisymmetric solitary waves, J. Phys. A, 15 (1982), 2599-2619. doi: 10.1088/0305-4470/15/8/036.  Google Scholar

[18]

L. Landau and E. Lifshitz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sow., 8 (1935), 153-169. Google Scholar

[19]

C. Lin and K. Wu, Singular limits of the Klein-Gordon equation, Arch. Rational Mech. Anal., 197 (2010), 689-711. doi: 10.1007/s00205-010-0324-8.  Google Scholar

[20]

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.  Google Scholar

[21]

F. H. Lin and J. Shatah, Soliton dynamics in planar ferromagnets and anti-ferromagnets, Journal of Zhejiang University Science, 4 (2003), 503-510. Google Scholar

[22]

F. Lin and J. Wei, Travelling wave solutions of Schrödinger map equation, Comm. Pure Appl. Math., 63 (2010), 1585-1621. doi: 10.1002/cpa.20338.  Google Scholar

[23]

T.-C. Lin, J. Wei and J. Yang, Vortex rings for the Gross-Pitaevskii equation in $\mathbb R^3$, J. Math. Pures Appl., 100 (2013), 69-112. doi: 10.1016/j.matpur.2012.10.012.  Google Scholar

[24]

J. C. Neu, Vortices in complex scalar fields, Phy. D, 43 (1990), 385-406. doi: 10.1016/0167-2789(90)90143-D.  Google Scholar

[25]

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

[26]

Y. Yu, Vortex dynamics for nonlinear Klein-Gordon equation, J. Differential Equations, 251 (2011), 970-994. doi: 10.1016/j.jde.2011.04.023.  Google Scholar

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