# American Institute of Mathematical Sciences

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

 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
 $A_+, A_->0,\ B^20,$
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.
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.

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##### 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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