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A variational approach to three-phase traveling waves for a gradient system
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 |
$ {\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*} $ |
$ A_+, A_->0,\ B^2<A_+A_-,\ t^+, t^->0, $ |
$ w(x) = (w^+, w^-): {\mathbb R}^2 \rightarrow\mathbb{C}^2 $ |
$ (1, 1) $ |
$ {\mathcal L} $ |
$ w $ |
$ B<0 $ |
$ {\mathcal L} $ |
$ \frac{\partial w}{\partial{x_1}} $ |
$ \frac{\partial w}{\partial{x_2}} $ |
$ {\mathcal L} $ |
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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