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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  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 & Continuous Dynamical Systems, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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, 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. 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, 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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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