American Institute of Mathematical Sciences

Advanced
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 & 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.  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

[1]

Hans G. Kaper, Peter Takáč. Bifurcating vortex solutions of the complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 871-880. doi: 10.3934/dcds.1999.5.871
[2]

Shijin Ding, Qiang Du. The global minimizers and vortex solutions to a Ginzburg-Landau model of superconducting films. Communications on Pure & Applied Analysis, 2002, 1 (3) : 327-340. doi: 10.3934/cpaa.2002.1.327
[3]

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
[4]

Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345
[5]

N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934/proc.2005.2005.476
[6]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507
[7]

Sen-Zhong Huang, Peter Takáč. Global smooth solutions of the complex Ginzburg-Landau equation and their dynamical properties. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 825-848. doi: 10.3934/dcds.1999.5.825
[8]

Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155
[9]

Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587
[10]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
[11]

Hong Lu, Mingji Zhang. Dynamics of non-autonomous fractional Ginzburg-Landau equations driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3553-3576. doi: 10.3934/dcdsb.2020072
[12]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233
[13]

Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934/cpaa.2019109
[14]

Feng Zhou, Chunyou Sun. Dynamics for the complex Ginzburg-Landau equation on non-cylindrical domains I: The diffeomorphism case. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3767-3792. doi: 10.3934/dcdsb.2016120
[15]

Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934/dcdsb.2014.19.1801
[16]

Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205
[17]

Mickaël Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934/nhm.2011.6.715
[18]

Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 121-142. doi: 10.3934/dcds.2000.6.121
[19]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[20]

Shujuan Lü, Chunbiao Gan, Baohua Wang, Linning Qian, Meisheng Li. Traveling wave solutions and its stability for 3D Ginzburg-Landau type equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 507-527. doi: 10.3934/dcdsb.2011.16.507

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (148)
  • HTML views (213)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy