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October  2021, 41(10): 4767-4790. doi: 10.3934/dcds.2021056

On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system

Lipeng Duan 1, and  Jun Yang 1,2,,

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
2. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

* Corresponding author: jyang2019@gzhu.edu.cn

Received  July 2020 Published  October 2021 Early access  March 2021

Fund Project: The authors are supported by the grants CCNU18CXTD04 and NSFC (No. 11771167 & No. 11831009)

For the coupled Ginzburg-Landau system in
$ {\mathbb R}^2 $
$ \begin{align*} \begin{cases} -\Delta w^+ +\Big[A_+\big(|w^+|^2-{t^+}^2\big)+B\big(|w^-|^2-{t^-}^2\big)\Big]w^+ = 0, \\ -\Delta w^- +\Big[A_-\big(|w^-|^2-{t^-}^2\big)+B\big(|w^+|^2-{t^+}^2\big)\Big]w^- = 0, \end{cases} \end{align*} $
with following constraints for the constant coefficients
$ A_+, A_->0,\ B^2<A_+A_-,\ t^+, t^->0, $
the radially symmetric solution
$ w(x) = (w^+, w^-): {\mathbb R}^2 \rightarrow\mathbb{C}^2 $
 of degree pair
$ (1, 1) $
 was given by A. Alama and Q. Gao in J. Differential Equations 255 (2013), 3564-3591. We will concern its linearized operator
$ {\mathcal L} $
 around
$ w $
 and prove the non-degeneracy result under one more assumption
$ B<0 $
: the kernel of
$ {\mathcal L} $
 is spanned by the functions
$ \frac{\partial w}{\partial{x_1}} $
 and
$ \frac{\partial w}{\partial{x_2}} $
 in a natural Hilbert space. As an application of the non-degeneracy result, a solvability theory for the linearized operator
$ {\mathcal L} $
 will be given.
Keywords: Non-degeneracy, Ginzburg-Landau system, superconductors, Fredholm alternative, vortex solutions.
Mathematics Subject Classification: Primary: 35B35, 35Q56.
Citation: Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4767-4790. doi: 10.3934/dcds.2021056
References:
[1]

S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model, J. Funct. Anal., 256 (2009), 1118-1136.  doi: 10.1016/j.jfa.2008.10.021.

[2]

S. Alama, L. Bronsard and P. Mironescu, On compound vortices in a two-component Ginzburg-Landau functional, Indiana Univ. Math. J., 61 (2012), 1861-1909.  doi: 10.1512/iumj.2012.61.4737.

[3]

S. Alama and Q. Gao, Symmetric vortices for two-component Ginzburg-Landau systems, J. Differential Equations, 255 (2013), 3564-3591.  doi: 10.1016/j.jde.2013.07.042.

[4]

S. Alama and Q. Gao, Stability of symmetric vortices for two-component Ginzburg-Landau systems, J. Funct. Anal., 267 (2014), 1751-1777.  doi: 10.1016/j.jfa.2014.06.013.

[5]

Y. Almog, L. Berlyand, D. Golovaty and I. Shafrir, Global minimizers for a $p$-Ginzburg-Landau-type energy in $\mathbb{R}^2$, J. Funct. Anal., 256 (2009), 2268-2290.  doi: 10.1016/j.jfa.2008.09.020.

[6]

Y. Almog, L. Berlyand, D. Golovaty and I. Shafrir, Radially symmetric minimizers for a p-Ginzburg Landau type energy in ${\mathbb R^2}$, Calc. Var. Partial Differential Equations, 42 (2011), 517-546.  doi: 10.1007/s00526-011-0396-9.

[7]

X. Chen, C. M. Elliott and T. Qi, 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.

[8]

M. Comte and P. Mironescu, A bifurcation analysis for the Ginzburg-Landau equation, Arch. Rational Mech. Anal., 144 (1998), 301-311.  doi: 10.1007/s002050050119.

[9]

J. Dávila, M. del Pino, M. Medina and R. Rodiac, Interacting helical vortex filaments in the 3-dimensional Ginzburg-Landau equation, preprint, arXiv: 1901.02807.

[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.

[11]

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.

[12]

S. Gustafson, Symmetric solutions of the Ginzburg Landau equation in all dimensions, Int. Math. Res. Not., 1997 (1997), 807-816.  doi: 10.1155/S1073792897000524.

[13]

R. Jiang, Y. Wang and J. Yang, Vortex structures for some geometric flows from pseudo-Euclidean spaces, Discrete Contin. Dyn. Syst., 39 (2019), 1745-1777.  doi: 10.3934/dcds.2019076.

[14]

K. Kasamatsu, M. Tsubota and M. Ueda, Structure of vortex lattices in rotating two-component Bose-Einstein condensates, Physica B, 329-333 (2004), 23-24.  doi: 10.1016/S0921-4526(02)01877-X.

[15]

A. Knigavko and B. Rosenstein, Spontaneous vortex state and ferromagnetic behavior of type-II $p$-wave superconductors, Physical Review B, 58 (1998), 9354-9364.  doi: 10.1103/PhysRevB.58.9354.

[16]

E. H. Lieb and M. Loss, Symmetry of the Ginzburg-Landau minimizer in a disc, Math. Res. Lett., 1 (1994), 701-715.  doi: 10.4310/MRL.1994.v1.n6.a7.

[17]

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

[18]

T.-C. Lin, The stability of the radial solution to the Ginzburg-Landau equation, Comm. Partial Differential Equations, 22 (1997), 619-632.  doi: 10.1080/03605309708821276.

[19]

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.

[20]

P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.  doi: 10.1006/jfan.1995.1073.

[21]

F. Pacard and T. Rivière, Linear and nonlinear aspects of vortices. (English summary) The Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, Birkhüser Boston, Inc., Boston, MA, 2000. doi: 10.1007/978-1-4612-1386-4.

[22]

M. Sauvageot, Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 375-397.  doi: 10.1007/s00030-003-0039-8.

[23]

J. Wei and J. Yang, Vortex rings pinning for the Gross-Pitaevskii equation in three dimensional space, SIAM J. Math. Anal., 44 (2012), 3991-4047.  doi: 10.1137/110860379.

[24]

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.

[25]

J. Yang, Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity, Discrete Contin. Dyn. Syst., 34 (2014), 2359-2388.  doi: 10.3934/dcds.2014.34.2359.

show all references

References:
[1]

S. Alama, L. Bronsard and P. Mironescu, On the structure of fractional degree vortices in a spinor Ginzburg-Landau model, J. Funct. Anal., 256 (2009), 1118-1136.  doi: 10.1016/j.jfa.2008.10.021.

[2]

S. Alama, L. Bronsard and P. Mironescu, On compound vortices in a two-component Ginzburg-Landau functional, Indiana Univ. Math. J., 61 (2012), 1861-1909.  doi: 10.1512/iumj.2012.61.4737.

[3]

S. Alama and Q. Gao, Symmetric vortices for two-component Ginzburg-Landau systems, J. Differential Equations, 255 (2013), 3564-3591.  doi: 10.1016/j.jde.2013.07.042.

[4]

S. Alama and Q. Gao, Stability of symmetric vortices for two-component Ginzburg-Landau systems, J. Funct. Anal., 267 (2014), 1751-1777.  doi: 10.1016/j.jfa.2014.06.013.

[5]

Y. Almog, L. Berlyand, D. Golovaty and I. Shafrir, Global minimizers for a $p$-Ginzburg-Landau-type energy in $\mathbb{R}^2$, J. Funct. Anal., 256 (2009), 2268-2290.  doi: 10.1016/j.jfa.2008.09.020.

[6]

Y. Almog, L. Berlyand, D. Golovaty and I. Shafrir, Radially symmetric minimizers for a p-Ginzburg Landau type energy in ${\mathbb R^2}$, Calc. Var. Partial Differential Equations, 42 (2011), 517-546.  doi: 10.1007/s00526-011-0396-9.

[7]

X. Chen, C. M. Elliott and T. Qi, 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.

[8]

M. Comte and P. Mironescu, A bifurcation analysis for the Ginzburg-Landau equation, Arch. Rational Mech. Anal., 144 (1998), 301-311.  doi: 10.1007/s002050050119.

[9]

J. Dávila, M. del Pino, M. Medina and R. Rodiac, Interacting helical vortex filaments in the 3-dimensional Ginzburg-Landau equation, preprint, arXiv: 1901.02807.

[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.

[11]

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.

[12]

S. Gustafson, Symmetric solutions of the Ginzburg Landau equation in all dimensions, Int. Math. Res. Not., 1997 (1997), 807-816.  doi: 10.1155/S1073792897000524.

[13]

R. Jiang, Y. Wang and J. Yang, Vortex structures for some geometric flows from pseudo-Euclidean spaces, Discrete Contin. Dyn. Syst., 39 (2019), 1745-1777.  doi: 10.3934/dcds.2019076.

[14]

K. Kasamatsu, M. Tsubota and M. Ueda, Structure of vortex lattices in rotating two-component Bose-Einstein condensates, Physica B, 329-333 (2004), 23-24.  doi: 10.1016/S0921-4526(02)01877-X.

[15]

A. Knigavko and B. Rosenstein, Spontaneous vortex state and ferromagnetic behavior of type-II $p$-wave superconductors, Physical Review B, 58 (1998), 9354-9364.  doi: 10.1103/PhysRevB.58.9354.

[16]

E. H. Lieb and M. Loss, Symmetry of the Ginzburg-Landau minimizer in a disc, Math. Res. Lett., 1 (1994), 701-715.  doi: 10.4310/MRL.1994.v1.n6.a7.

[17]

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

[18]

T.-C. Lin, The stability of the radial solution to the Ginzburg-Landau equation, Comm. Partial Differential Equations, 22 (1997), 619-632.  doi: 10.1080/03605309708821276.

[19]

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.

[20]

P. Mironescu, On the stability of radial solutions of the Ginzburg-Landau equation, J. Funct. Anal., 130 (1995), 334-344.  doi: 10.1006/jfan.1995.1073.

[21]

F. Pacard and T. Rivière, Linear and nonlinear aspects of vortices. (English summary) The Ginzburg-Landau model. Progress in Nonlinear Differential Equations and their Applications, Birkhüser Boston, Inc., Boston, MA, 2000. doi: 10.1007/978-1-4612-1386-4.

[22]

M. Sauvageot, Properties of the solutions of the Ginzburg-Landau equation on the bifurcation branch, NoDEA Nonlinear Differential Equations Appl., 10 (2003), 375-397.  doi: 10.1007/s00030-003-0039-8.

[23]

J. Wei and J. Yang, Vortex rings pinning for the Gross-Pitaevskii equation in three dimensional space, SIAM J. Math. Anal., 44 (2012), 3991-4047.  doi: 10.1137/110860379.

[24]

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.

[25]

J. Yang, Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity, Discrete Contin. Dyn. Syst., 34 (2014), 2359-2388.  doi: 10.3934/dcds.2014.34.2359.

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