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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 - A, 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. 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, (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, (2008), 55. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. Google Scholar

[19]

C. Lin and K. Wu, Singular limits of the Klein-Gordon equation,, Arch. Rational Mech. Anal., 197 (2010), 689. 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. 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. Google Scholar

[22]

F. Lin and J. Wei, Travelling wave solutions of Schrödinger map equation,, Comm. Pure Appl. Math., 63 (2010), 1585. 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. doi: 10.1016/j.matpur.2012.10.012. Google Scholar

[24]

J. C. Neu, Vortices in complex scalar fields,, Phy. D, 43 (1990), 385. 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. 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. 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. 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, (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, (2008), 55. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (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. 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. Google Scholar

[19]

C. Lin and K. Wu, Singular limits of the Klein-Gordon equation,, Arch. Rational Mech. Anal., 197 (2010), 689. 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. 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. Google Scholar

[22]

F. Lin and J. Wei, Travelling wave solutions of Schrödinger map equation,, Comm. Pure Appl. Math., 63 (2010), 1585. 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. doi: 10.1016/j.matpur.2012.10.012. Google Scholar

[24]

J. C. Neu, Vortices in complex scalar fields,, Phy. D, 43 (1990), 385. 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. 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. doi: 10.1016/j.jde.2011.04.023. Google Scholar

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