
-
Previous Article
Positive solutions for a nonlinear Schrödinger-Poisson system
- DCDS Home
- This Issue
-
Next Article
On a new two-component $b$-family peakon system with cubic nonlinearity
Periodic solutions for the N-vortex problem via a superposition principle
Mathematisches Institut, Universität Gießen, Arndstr. 2, 35392 Gießen, Germany |
$N$ |
$Ω\subset\mathbb{R}^2$ |
$Γ_k\dot{z}_k = J\nabla_{z_k}H(z_1,...,z_N),\ \ \ \ k = 1,...,N,$ |
$Γ_k∈\mathbb{R}\setminus\{0\}$ |
$k$ |
$z_k(t)∈Ω$ |
$J∈\mathbb{R}^{2× 2}$ |
$H(z_1,...,z_N) = -\frac{1}{2π}\sum\limits_{\underset{k≠ j}{k,j = 1}}^NΓ_jΓ_k\log|z_k-z_j|-\sum\limits_{k,j = 1}^NΓ_jΓ_k g(z_k,z_j)$ |
$g:Ω×Ω \to \mathbb{R}$ |
$N$ |
$T$ |
$T>0$ |
$Ω = B_1(0)$ |
$B_1(0)$ |
References:
[1] |
H. Aref,
Relative equilibria of point vortices and the fundamental theorem of algebra, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2168-2184.
doi: 10.1098/rspa.2010.0580. |
[2] |
H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda and D. L. Vainchtein,
Vortex crystals, Adv. Appl. Mech., 39 (2003), 1-79.
doi: 10.1016/S0065-2156(02)39001-X. |
[3] |
T. Bartsch,
A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 691-718.
doi: 10.1016/S0294-1449(97)80130-8. |
[4] |
T. Bartsch and Q. Dai,
Periodic solutions of the N-vortex Hamiltonian system in planar domains, J. Diff. Equ., 260 (2016), 2275-2295.
doi: 10.1016/j.jde.2015.10.002. |
[5] |
T. Bartsch, Q. Dai and B. Gebhard,
Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary, SIAM J. Appl. Math., 78 (2018), 977-995.
doi: 10.1137/16M1107085. |
[6] |
T. Bartsch and B. Gebhard,
Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Math. Ann., 369 (2017), 627-651.
doi: 10.1007/s00208-016-1505-z. |
[7] |
T. Bartsch, A. M. Micheletti and A. Pistoia, The Morse property for functions of Kirchhoff-Routh path type, to appear: Discrete Contin. Dyn. Syst. Ser. S. |
[8] |
T. Bartsch and A. Pistoia,
Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75 (2015), 726-744.
doi: 10.1137/140981253. |
[9] |
T. Bartsch, A. Pistoia and T. Weth,
N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), 653-686.
doi: 10.1007/s00220-010-1053-4. |
[10] |
T. Bartsch and M. Sacchet,
Periodic solutions with prescribed minimal period of vortex type problems in domains, Nonlinearity, 31 (2018), 2156-2172.
doi: 10.1088/1361-6544/aaaf2d. |
[11] |
D. Cao, Z. Liu and J. Wei,
Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Rat. Mech. Anal., 212 (2014), 179-217.
doi: 10.1007/s00205-013-0692-y. |
[12] |
O. Cornea, G. Lupton, J. Oprea and D. Tanré,
Lusternik-Schnirelmann Category, American Mathematical Society, Providence Rhode Island, 2003.
doi: 10.1090/surv/103. |
[13] |
M. del Pino, M. Kowalczyk and M. Musso,
Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81.
doi: 10.1007/s00526-004-0314-5. |
[14] |
P. Esposito, M. Grossi and A. Pistoia,
On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.
doi: 10.1016/j.anihpc.2004.12.001. |
[15] |
M. Flucher, Variational Problems with Concentration, Birkhäuser, Basel Boston Berlin, 1999.
doi: 10.1007/978-3-0348-8687-1. |
[16] |
M. Gelantalis and P. Sternberg, Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on $ S^2$,
J. Math. Phys., 53 (2012), 083701, 24pp.
doi: 10.1063/1.4739748. |
[17] |
R. L. Jerrard and H. M. Soner,
Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99-125.
doi: 10.1007/s002050050085. |
[18] |
K. Khanin,
Quasi-periodic motions of vortex systems, Phys. D, 4 (1982), 261-269.
doi: 10.1016/0167-2789(82)90067-7. |
[19] |
G. R. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876. |
[20] |
C. Kuhl,
Symmetric equilibria for the N-vortex problem, J. Fixed Point Theory Appl., 17 (2015), 597-624.
doi: 10.1007/s11784-015-0242-3. |
[21] |
C. Kuhl,
Equilibria for the N-vortex-problem in a general bounded domain, J. Math. Anal. Appl., 433 (2016), 1531-1560.
doi: 10.1016/j.jmaa.2015.08.055. |
[22] |
M. Kurzke, C. Melcher, R. Moser and D. Spirn,
Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation, Arch. Rat. Mech. Anal., 199 (2011), 843-888.
doi: 10.1007/s00205-010-0356-0. |
[23] |
C. C. Lin,
On the motion of vortices in two dimensions i. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 570-575.
|
[24] |
C. C. Lin,
On the motion of vortices in two dimensions ii. Some further investigations on the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 575-577.
|
[25] |
C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer, Berlin Heidelberg, 1984. |
[26] |
C. Marchioro and M. Pulvirenti,
Mathematical Theory of Incompressible Nonviscous Fluids, Springer, New York, 1994.
doi: 10.1007/978-1-4612-4284-0. |
[27] |
J. Moser,
Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747.
doi: 10.1002/cpa.3160290613. |
[28] |
P. K. Newton,
The N-Vortex Problem, Springer, New York, 2001.
doi: 10.1007/978-1-4684-9290-3. |
[29] |
G. E. Roberts,
Stability of relative equilibria in the planar N-vortex problem, SIAM J. Appl. Dyn. Sys., 12 (2013), 1114-1134.
doi: 10.1137/130907434. |
[30] |
E. J. Routh,
Some applications of conjugate functions, Proc. London Math. Soc., 12 (1880), 73-89.
doi: 10.1112/plms/s1-12.1.73. |
[31] |
P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. |
[32] |
R. Venkatraman, Periodic orbits of Gross-Pitaevskii in the disc with vortices following point vortex flow, Calc. Var. Partial Differential Equations, 56 (2017), Art. 64, 35 pp.
doi: 10.1007/s00526-017-1168-y. |
[33] |
A. Weinstein,
Normal modes for nonlinear Hamiltonian systems, Inventiones mathematicae, 20 (1973), 47-57.
doi: 10.1007/BF01405263. |
show all references
References:
[1] |
H. Aref,
Relative equilibria of point vortices and the fundamental theorem of algebra, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 2168-2184.
doi: 10.1098/rspa.2010.0580. |
[2] |
H. Aref, P. K. Newton, M. A. Stremler, T. Tokieda and D. L. Vainchtein,
Vortex crystals, Adv. Appl. Mech., 39 (2003), 1-79.
doi: 10.1016/S0065-2156(02)39001-X. |
[3] |
T. Bartsch,
A generalization of the Weinstein-Moser theorems on periodic orbits of a Hamiltonian system near an equilibrium, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 691-718.
doi: 10.1016/S0294-1449(97)80130-8. |
[4] |
T. Bartsch and Q. Dai,
Periodic solutions of the N-vortex Hamiltonian system in planar domains, J. Diff. Equ., 260 (2016), 2275-2295.
doi: 10.1016/j.jde.2015.10.002. |
[5] |
T. Bartsch, Q. Dai and B. Gebhard,
Periodic solutions of N-vortex type Hamiltonian systems near the domain boundary, SIAM J. Appl. Math., 78 (2018), 977-995.
doi: 10.1137/16M1107085. |
[6] |
T. Bartsch and B. Gebhard,
Global continua of periodic solutions of singular first-order Hamiltonian systems of N-vortex type, Math. Ann., 369 (2017), 627-651.
doi: 10.1007/s00208-016-1505-z. |
[7] |
T. Bartsch, A. M. Micheletti and A. Pistoia, The Morse property for functions of Kirchhoff-Routh path type, to appear: Discrete Contin. Dyn. Syst. Ser. S. |
[8] |
T. Bartsch and A. Pistoia,
Critical points of the N-vortex Hamiltonian in bounded planar domains and steady state solutions of the incompressible Euler equations, SIAM J. Appl. Math., 75 (2015), 726-744.
doi: 10.1137/140981253. |
[9] |
T. Bartsch, A. Pistoia and T. Weth,
N-vortex equilibria for ideal fluids in bounded planar domains and new nodal solutions of the sinh-poisson and the Lane-Emden-Fowler equations, Comm. Math. Phys., 297 (2010), 653-686.
doi: 10.1007/s00220-010-1053-4. |
[10] |
T. Bartsch and M. Sacchet,
Periodic solutions with prescribed minimal period of vortex type problems in domains, Nonlinearity, 31 (2018), 2156-2172.
doi: 10.1088/1361-6544/aaaf2d. |
[11] |
D. Cao, Z. Liu and J. Wei,
Regularization of point vortices pairs for the Euler equation in dimension two, Arch. Rat. Mech. Anal., 212 (2014), 179-217.
doi: 10.1007/s00205-013-0692-y. |
[12] |
O. Cornea, G. Lupton, J. Oprea and D. Tanré,
Lusternik-Schnirelmann Category, American Mathematical Society, Providence Rhode Island, 2003.
doi: 10.1090/surv/103. |
[13] |
M. del Pino, M. Kowalczyk and M. Musso,
Singular limits in Liouville-type equations, Calc. Var. Partial Differential Equations, 24 (2005), 47-81.
doi: 10.1007/s00526-004-0314-5. |
[14] |
P. Esposito, M. Grossi and A. Pistoia,
On the existence of blowing-up solutions for a mean field equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 227-257.
doi: 10.1016/j.anihpc.2004.12.001. |
[15] |
M. Flucher, Variational Problems with Concentration, Birkhäuser, Basel Boston Berlin, 1999.
doi: 10.1007/978-3-0348-8687-1. |
[16] |
M. Gelantalis and P. Sternberg, Rotating 2N-vortex solutions to the Gross-Pitaevskii equation on $ S^2$,
J. Math. Phys., 53 (2012), 083701, 24pp.
doi: 10.1063/1.4739748. |
[17] |
R. L. Jerrard and H. M. Soner,
Dynamics of Ginzburg-Landau vortices, Arch. Rat. Mech. Anal., 142 (1998), 99-125.
doi: 10.1007/s002050050085. |
[18] |
K. Khanin,
Quasi-periodic motions of vortex systems, Phys. D, 4 (1982), 261-269.
doi: 10.1016/0167-2789(82)90067-7. |
[19] |
G. R. Kirchhoff, Vorlesungen Über Mathematische Physik, Teubner, Leipzig, 1876. |
[20] |
C. Kuhl,
Symmetric equilibria for the N-vortex problem, J. Fixed Point Theory Appl., 17 (2015), 597-624.
doi: 10.1007/s11784-015-0242-3. |
[21] |
C. Kuhl,
Equilibria for the N-vortex-problem in a general bounded domain, J. Math. Anal. Appl., 433 (2016), 1531-1560.
doi: 10.1016/j.jmaa.2015.08.055. |
[22] |
M. Kurzke, C. Melcher, R. Moser and D. Spirn,
Ginzburg-Landau vortices driven by the Landau-Lifshitz-Gilbert equation, Arch. Rat. Mech. Anal., 199 (2011), 843-888.
doi: 10.1007/s00205-010-0356-0. |
[23] |
C. C. Lin,
On the motion of vortices in two dimensions i. Existence of the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 570-575.
|
[24] |
C. C. Lin,
On the motion of vortices in two dimensions ii. Some further investigations on the Kirchhoff-Routh function, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 575-577.
|
[25] |
C. Marchioro and M. Pulvirenti, Vortex Methods in Two-Dimensional Fluid Dynamics, Springer, Berlin Heidelberg, 1984. |
[26] |
C. Marchioro and M. Pulvirenti,
Mathematical Theory of Incompressible Nonviscous Fluids, Springer, New York, 1994.
doi: 10.1007/978-1-4612-4284-0. |
[27] |
J. Moser,
Periodic orbits near an equilibrium and a theorem by Alan Weinstein, Comm. Pure Appl. Math., 29 (1976), 727-747.
doi: 10.1002/cpa.3160290613. |
[28] |
P. K. Newton,
The N-Vortex Problem, Springer, New York, 2001.
doi: 10.1007/978-1-4684-9290-3. |
[29] |
G. E. Roberts,
Stability of relative equilibria in the planar N-vortex problem, SIAM J. Appl. Dyn. Sys., 12 (2013), 1114-1134.
doi: 10.1137/130907434. |
[30] |
E. J. Routh,
Some applications of conjugate functions, Proc. London Math. Soc., 12 (1880), 73-89.
doi: 10.1112/plms/s1-12.1.73. |
[31] |
P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992. |
[32] |
R. Venkatraman, Periodic orbits of Gross-Pitaevskii in the disc with vortices following point vortex flow, Calc. Var. Partial Differential Equations, 56 (2017), Art. 64, 35 pp.
doi: 10.1007/s00526-017-1168-y. |
[33] |
A. Weinstein,
Normal modes for nonlinear Hamiltonian systems, Inventiones mathematicae, 20 (1973), 47-57.
doi: 10.1007/BF01405263. |

[1] |
J. R. Ward. Periodic solutions of first order systems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 381-389. doi: 10.3934/dcds.2013.33.381 |
[2] |
Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064 |
[3] |
Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807 |
[4] |
Luis Vega. The dynamics of vortex filaments with corners. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1581-1601. doi: 10.3934/cpaa.2015.14.1581 |
[5] |
Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116 |
[6] |
Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222 |
[7] |
Xavier Perrot, Xavier Carton. Point-vortex interaction in an oscillatory deformation field: Hamiltonian dynamics, harmonic resonance and transition to chaos. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 971-995. doi: 10.3934/dcdsb.2009.11.971 |
[8] |
Juhong Kuang, Weiyi Chen, Zhiming Guo. Periodic solutions with prescribed minimal period for second order even Hamiltonian systems. Communications on Pure and Applied Analysis, 2022, 21 (1) : 47-59. doi: 10.3934/cpaa.2021166 |
[9] |
Chjan C. Lim, Ka Kit Tung. Introduction: Recent advances in vortex dynamics and turbulence. Discrete and Continuous Dynamical Systems - B, 2005, 5 (1) : i-i. doi: 10.3934/dcdsb.2005.5.1i |
[10] |
Carlos García-Azpeitia. Relative periodic solutions of the $ n $-vortex problem on the sphere. Journal of Geometric Mechanics, 2019, 11 (3) : 427-438. doi: 10.3934/jgm.2019021 |
[11] |
Xiangjin Xu. Sub-harmonics of first order Hamiltonian systems and their asymptotic behaviors. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 643-654. doi: 10.3934/dcdsb.2003.3.643 |
[12] |
Cyril Joel Batkam. Homoclinic orbits of first-order superquadratic Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3353-3369. doi: 10.3934/dcds.2014.34.3353 |
[13] |
Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure and Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75 |
[14] |
Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065 |
[15] |
Zhongyi Huang, Peter A. Markowich, Christof Sparber. Numerical simulation of trapped dipolar quantum gases: Collapse studies and vortex dynamics. Kinetic and Related Models, 2010, 3 (1) : 181-194. doi: 10.3934/krm.2010.3.181 |
[16] |
Pedro J. Torres, R. Carretero-González, S. Middelkamp, P. Schmelcher, Dimitri J. Frantzeskakis, P.G. Kevrekidis. Vortex interaction dynamics in trapped Bose-Einstein condensates. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1589-1615. doi: 10.3934/cpaa.2011.10.1589 |
[17] |
Zhiguo Xu, Weizhu Bao, Shaoyun Shi. Quantized vortex dynamics and interaction patterns in superconductivity based on the reduced dynamical law. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2265-2297. doi: 10.3934/dcdsb.2018096 |
[18] |
Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218 |
[19] |
Tianqing An, Zhi-Qiang Wang. Periodic solutions of Hamiltonian systems with anisotropic growth. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1069-1082. doi: 10.3934/cpaa.2010.9.1069 |
[20] |
Alessandro Fonda, Andrea Sfecci. Multiple periodic solutions of Hamiltonian systems confined in a box. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1425-1436. doi: 10.3934/dcds.2017059 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]