• Previous Article
    The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times
  • DCDS-B Home
  • This Issue
  • Next Article
    Dynamics of fractional nonclassical diffusion equations with delay driven by additive noise on $ \mathbb{R}^n $
doi: 10.3934/dcdsb.2021234
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus

Department of Applied Mathematics, National Yang Ming Chiao Tung University, Taiwan, National Center for Theoretical Sciences, National Taiwan University, Taipei, Taiwan

* Corresponding author: Hsin-Yuan Huang

Received  January 2021 Revised  July 2021 Early access September 2021

Fund Project: The author is partially supported by Ministry of Science and Technology of Taiwan under the grant MOST 106-2628-M-009-005-MY4

In this paper, we study an elliptic system arising from the U(1)
$ \times $
U(1) Abelian Chern-Simons Model[25,37] of the form
$ \begin{equation} \left\{\begin{split} \Delta u = &\lambda \left(a(b-a)e^{u}-b(b-a)e^{v}+a^2e^{2u} -abe^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_1}m_j\delta_{p_j}, \\ \Delta v = &\lambda \left(-b(b-a)e^{u}+a(b-a)e^{v}-abe^{2u} +a^2e^{2v}+b(b-a)e^{u+v}\right)\\ & +4\pi \sum\limits_{j = 1}^{k_2}n_j\delta_{q_j}, \end{split}\right. \quad\quad\quad\quad (1)\end{equation} $
which are defined on a parallelogram
$ \Omega $
in
$ \mathbb{R}^2 $
with doubly periodic boundary conditions. Here,
$ a $
and
$ b $
are interaction constants,
$ \lambda>0 $
is related to coupling constant,
$ m_j>0(j = 1,\cdots,k_1) $
,
$ n_j>0(j = 1,\cdots,k_2) $
,
$ \delta_{p} $
is the Dirac measure,
$ p $
is called vortex point. Concerning the existence results of this system over
$ \Omega $
, only the cases
$ (a,b) = (0,1) $
[28] and
$ a>b>0 $
[14] were studied in the literature. The solvability of this system (1) is still an open problem as regards other parameters
$ (a,b) $
. We show that the system (1) admits topological solutions provided
$ \lambda $
is large and
$ b>a>0 $
Our arguments are based on a iteration scheme and variational formulation.
Citation: Hsin-Yuan Huang. Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021234
References:
[1]

A. A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solids, 2 (1957), 199-208.  doi: 10.1016/0022-3697(57)90083-5.  Google Scholar

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, volume 252 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar

[3]

E. B. Bogomol'n$\check{\mathrm{y}}$, The stability of classical solutions, Jadernaja Fiz., 24 (1976), 861-870.   Google Scholar

[4]

L. A. Caffarelli and Y. Yang, Vortex condensation in the Chern-Simons Higgs model: An existence theorem, Comm. Math. Phys., 168 (1995), 321-336.  doi: 10.1007/BF02101552.  Google Scholar

[5]

C.-C. ChenC.-S. Lin and G. Wang, Concentration phenomena of two-vortex solutions in a Chern-Simons model, Ann. Sc. Norm. Super. Pisa., 3 (2004), 367-397.   Google Scholar

[6]

J.-L. ChernZ.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys., 296 (2010), 323-351.  doi: 10.1007/s00220-010-1021-z.  Google Scholar

[7]

K. Choe, Uniqueness of the topological multivortex solution in the self-dual Chern-Simons theory, J. Math. Phys., 46 (2005), 012305, 22 pp. doi: 10.1063/1.1834694.  Google Scholar

[8]

K. ChoeN. Kim and C.-S. Lin, Existence of radial mixed type solutions in Chern–Simons theories of rank 2 in $\mathbb{R}^2$, Comm. Math. Phys., 370 (2019), 995-1017.  doi: 10.1007/s00220-019-03469-6.  Google Scholar

[9]

M. del PinoP. EspositoP. Figueroa and M. Musso, Nontopological condensates for the self-dual Chern-Simons-Higgs model, Comm. Pure Appl. Math., 68 (2015), 1191-1283.  doi: 10.1002/cpa.21548.  Google Scholar

[10]

J. Dziarmaga, Low energy dynamics of ${[\mathrm{U}(1)]}^{N}$ chern-simons solitons, Phys. Rev. D, 49 (1994), 5469-5479.   Google Scholar

[11]

J. Fröhlich and P. A. Marchetti, Quantum field theory of anyons, Lett. Math. Phys., 16 (1988), 347-358.  doi: 10.1007/BF00402043.  Google Scholar

[12]

J. Fröhlich and P. A. Marchetti, Quantum field theories of vortices and anyons, Comm. Math. Phys., 121 (1989), 177-223.  doi: 10.1007/BF01217803.  Google Scholar

[13]

X. HanH.-Y. Huang and C.-S. Lin, Bubbling solutions for a skew-symmetric Chern-Simons system in a torus, J. Funct. Anal., 273 (2017), 1354-1396.  doi: 10.1016/j.jfa.2017.04.018.  Google Scholar

[14]

X. Han and G. Tarantello, Doubly periodic self-dual vortices in a relativistic non-abelian Chern–Simons model, Calc. Var. Partial Differ. Equ., 49 (2014), 1149-1176.  doi: 10.1007/s00526-013-0615-7.  Google Scholar

[15]

J. HongY. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.  Google Scholar

[16]

G. Huang and C.-S. Lin, The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.  doi: 10.1512/iumj.2016.65.5769.  Google Scholar

[17]

H.-Y. Huang, Y. Lee and C.-S. Lin, Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system, J. Math. Phys., 56 (2015), 041501, 12 pp. doi: 10.1063/1.4916290.  Google Scholar

[18]

H.-Y. Huang and C.-S. Lin, Uniqueness of non-topological solutions for the Chern-Simons system with two Higgs particles, Kodai Math. J., 37 (2014), 274-284.  doi: 10.2996/kmj/1404393887.  Google Scholar

[19]

H.-Y. Huang and C.-S. Lin, Classification of the entire radial self-dual solutions to non-abelian Chern–Simons systems, J. Funct. Anal., 266 (2014), 6796-6841.  doi: 10.1016/j.jfa.2014.03.007.  Google Scholar

[20]

H.-Y. Huang and L. Zhang, The domain geometry and the bubbling phenomenon of rank two gauge theory, Comm. Math. Phys., 349 (2017), 393-424.  doi: 10.1007/s00220-016-2685-9.  Google Scholar

[21]

R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234.  Google Scholar

[22]

A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Birkhäuser, Boston, Mass., 1980.  Google Scholar

[23]

B. Julia and A. Zee, Poles with both magnetic and electric charges in non-abelian gauge theory, Physical Review D, 11 (1975), 2227.  doi: 10.1103/PhysRevD.11.2227.  Google Scholar

[24]

D. I. Khomskii and A. Freimuth, Charged vortices in high temperature superconductors, Physical Review Letters, 75 (1995), 1384.  doi: 10.1103/PhysRevLett.75.1384.  Google Scholar

[25]

C. KimC. LeeP. KoB.-H. Lee and H. Min, Schrödinger fields on the plane with ${[\mathrm{U}(1)]}^{N}$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840.  doi: 10.1103/PhysRevD.48.1821.  Google Scholar

[26]

K. KumagaiK. Nozaki and Y. Matsuda, Charged vortices in high-temperature superconductors probed by NMR, Phys. Rev. B, 63 (2001), 144502.  doi: 10.1103/PhysRevB.63.144502.  Google Scholar

[27]

C.-S. LinA. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350.  doi: 10.1016/j.jfa.2007.03.010.  Google Scholar

[28]

C.-S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347.  doi: 10.1007/s00220-009-0774-8.  Google Scholar

[29]

C.-S. Lin and S. Yan, Existence of bubbling solutions for Chern-Simons model on a torus, Arc. Ration. Mech. Anal., 207 (2013), 353-392.  doi: 10.1007/s00205-012-0575-7.  Google Scholar

[30]

C.-S. Lin and S. Yan, On condensate of solutions for the Chern-Simons-Higgs equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1329-1354.  doi: 10.1016/j.anihpc.2016.10.006.  Google Scholar

[31]

C.-S. Lin and S. Yan, On the mean field type bubbling solutions for Chern-Simons-Higgs equation, Adv. Math., 338 (2018), 1141-1188.  doi: 10.1016/j.aim.2018.09.021.  Google Scholar

[32]

A. Poliakovsky and G. Tarantello, On non-topological solutions for planar liouville systems of Toda-type, Comm. Math. Phys., 347 (2016), 223-270.  doi: 10.1007/s00220-016-2662-3.  Google Scholar

[33]

M. Prasad and C. Sommerfield, Exact classical solution for the't hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett., 35 (1975), 760.   Google Scholar

[34] L. H. Ryder, Quantum Field Theory, Cambridge university press, 1996.  doi: 10.1017/CBO9780511813900.  Google Scholar
[35]

J. Spruck and Y. Yang, The existence of nontopological solitons in the self-dual Chern-Simons theory, Comm. Math. Phys., 149 (1992), 361-376.  doi: 10.1007/BF02097630.  Google Scholar

[36]

G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9.  Google Scholar

[37]

F. Wilczek, Disassembling anyons, Phys. Rev. Lett., 69 (1992), 132-135.  doi: 10.1103/PhysRevLett.69.132.  Google Scholar

show all references

References:
[1]

A. A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solids, 2 (1957), 199-208.  doi: 10.1016/0022-3697(57)90083-5.  Google Scholar

[2]

T. Aubin, Nonlinear Analysis on Manifolds. Monge-Ampère Equations, volume 252 of Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4612-5734-9.  Google Scholar

[3]

E. B. Bogomol'n$\check{\mathrm{y}}$, The stability of classical solutions, Jadernaja Fiz., 24 (1976), 861-870.   Google Scholar

[4]

L. A. Caffarelli and Y. Yang, Vortex condensation in the Chern-Simons Higgs model: An existence theorem, Comm. Math. Phys., 168 (1995), 321-336.  doi: 10.1007/BF02101552.  Google Scholar

[5]

C.-C. ChenC.-S. Lin and G. Wang, Concentration phenomena of two-vortex solutions in a Chern-Simons model, Ann. Sc. Norm. Super. Pisa., 3 (2004), 367-397.   Google Scholar

[6]

J.-L. ChernZ.-Y. Chen and C.-S. Lin, Uniqueness of topological solutions and the structure of solutions for the Chern-Simons system with two Higgs particles, Comm. Math. Phys., 296 (2010), 323-351.  doi: 10.1007/s00220-010-1021-z.  Google Scholar

[7]

K. Choe, Uniqueness of the topological multivortex solution in the self-dual Chern-Simons theory, J. Math. Phys., 46 (2005), 012305, 22 pp. doi: 10.1063/1.1834694.  Google Scholar

[8]

K. ChoeN. Kim and C.-S. Lin, Existence of radial mixed type solutions in Chern–Simons theories of rank 2 in $\mathbb{R}^2$, Comm. Math. Phys., 370 (2019), 995-1017.  doi: 10.1007/s00220-019-03469-6.  Google Scholar

[9]

M. del PinoP. EspositoP. Figueroa and M. Musso, Nontopological condensates for the self-dual Chern-Simons-Higgs model, Comm. Pure Appl. Math., 68 (2015), 1191-1283.  doi: 10.1002/cpa.21548.  Google Scholar

[10]

J. Dziarmaga, Low energy dynamics of ${[\mathrm{U}(1)]}^{N}$ chern-simons solitons, Phys. Rev. D, 49 (1994), 5469-5479.   Google Scholar

[11]

J. Fröhlich and P. A. Marchetti, Quantum field theory of anyons, Lett. Math. Phys., 16 (1988), 347-358.  doi: 10.1007/BF00402043.  Google Scholar

[12]

J. Fröhlich and P. A. Marchetti, Quantum field theories of vortices and anyons, Comm. Math. Phys., 121 (1989), 177-223.  doi: 10.1007/BF01217803.  Google Scholar

[13]

X. HanH.-Y. Huang and C.-S. Lin, Bubbling solutions for a skew-symmetric Chern-Simons system in a torus, J. Funct. Anal., 273 (2017), 1354-1396.  doi: 10.1016/j.jfa.2017.04.018.  Google Scholar

[14]

X. Han and G. Tarantello, Doubly periodic self-dual vortices in a relativistic non-abelian Chern–Simons model, Calc. Var. Partial Differ. Equ., 49 (2014), 1149-1176.  doi: 10.1007/s00526-013-0615-7.  Google Scholar

[15]

J. HongY. Kim and P. Y. Pac, Multivortex solutions of the abelian Chern-Simons-Higgs theory, Phys. Rev. Lett., 64 (1990), 2230-2233.  doi: 10.1103/PhysRevLett.64.2230.  Google Scholar

[16]

G. Huang and C.-S. Lin, The existence of non-topological solutions for a skew-symmetric Chern-Simons system, Indiana Univ. Math. J., 65 (2016), 453-491.  doi: 10.1512/iumj.2016.65.5769.  Google Scholar

[17]

H.-Y. Huang, Y. Lee and C.-S. Lin, Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system, J. Math. Phys., 56 (2015), 041501, 12 pp. doi: 10.1063/1.4916290.  Google Scholar

[18]

H.-Y. Huang and C.-S. Lin, Uniqueness of non-topological solutions for the Chern-Simons system with two Higgs particles, Kodai Math. J., 37 (2014), 274-284.  doi: 10.2996/kmj/1404393887.  Google Scholar

[19]

H.-Y. Huang and C.-S. Lin, Classification of the entire radial self-dual solutions to non-abelian Chern–Simons systems, J. Funct. Anal., 266 (2014), 6796-6841.  doi: 10.1016/j.jfa.2014.03.007.  Google Scholar

[20]

H.-Y. Huang and L. Zhang, The domain geometry and the bubbling phenomenon of rank two gauge theory, Comm. Math. Phys., 349 (2017), 393-424.  doi: 10.1007/s00220-016-2685-9.  Google Scholar

[21]

R. Jackiw and E. J. Weinberg, Self-dual Chern-Simons vortices, Phys. Rev. Lett., 64 (1990), 2234-2237.  doi: 10.1103/PhysRevLett.64.2234.  Google Scholar

[22]

A. Jaffe and C. Taubes, Vortices and Monopoles: Structure of Static Gauge Theories, Birkhäuser, Boston, Mass., 1980.  Google Scholar

[23]

B. Julia and A. Zee, Poles with both magnetic and electric charges in non-abelian gauge theory, Physical Review D, 11 (1975), 2227.  doi: 10.1103/PhysRevD.11.2227.  Google Scholar

[24]

D. I. Khomskii and A. Freimuth, Charged vortices in high temperature superconductors, Physical Review Letters, 75 (1995), 1384.  doi: 10.1103/PhysRevLett.75.1384.  Google Scholar

[25]

C. KimC. LeeP. KoB.-H. Lee and H. Min, Schrödinger fields on the plane with ${[\mathrm{U}(1)]}^{N}$ Chern-Simons interactions and generalized self-dual solitons, Phys. Rev. D, 48 (1993), 1821-1840.  doi: 10.1103/PhysRevD.48.1821.  Google Scholar

[26]

K. KumagaiK. Nozaki and Y. Matsuda, Charged vortices in high-temperature superconductors probed by NMR, Phys. Rev. B, 63 (2001), 144502.  doi: 10.1103/PhysRevB.63.144502.  Google Scholar

[27]

C.-S. LinA. C. Ponce and Y. Yang, A system of elliptic equations arising in Chern-Simons field theory, J. Funct. Anal., 247 (2007), 289-350.  doi: 10.1016/j.jfa.2007.03.010.  Google Scholar

[28]

C.-S. Lin and J. V. Prajapat, Vortex condensates for relativistic abelian Chern-Simons model with two Higgs scalar fields and two gauge fields on a torus, Comm. Math. Phys., 288 (2009), 311-347.  doi: 10.1007/s00220-009-0774-8.  Google Scholar

[29]

C.-S. Lin and S. Yan, Existence of bubbling solutions for Chern-Simons model on a torus, Arc. Ration. Mech. Anal., 207 (2013), 353-392.  doi: 10.1007/s00205-012-0575-7.  Google Scholar

[30]

C.-S. Lin and S. Yan, On condensate of solutions for the Chern-Simons-Higgs equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1329-1354.  doi: 10.1016/j.anihpc.2016.10.006.  Google Scholar

[31]

C.-S. Lin and S. Yan, On the mean field type bubbling solutions for Chern-Simons-Higgs equation, Adv. Math., 338 (2018), 1141-1188.  doi: 10.1016/j.aim.2018.09.021.  Google Scholar

[32]

A. Poliakovsky and G. Tarantello, On non-topological solutions for planar liouville systems of Toda-type, Comm. Math. Phys., 347 (2016), 223-270.  doi: 10.1007/s00220-016-2662-3.  Google Scholar

[33]

M. Prasad and C. Sommerfield, Exact classical solution for the't hooft monopole and the Julia-Zee dyon, Phys. Rev. Lett., 35 (1975), 760.   Google Scholar

[34] L. H. Ryder, Quantum Field Theory, Cambridge university press, 1996.  doi: 10.1017/CBO9780511813900.  Google Scholar
[35]

J. Spruck and Y. Yang, The existence of nontopological solitons in the self-dual Chern-Simons theory, Comm. Math. Phys., 149 (1992), 361-376.  doi: 10.1007/BF02097630.  Google Scholar

[36]

G. Tarantello, Uniqueness of selfdual periodic Chern-Simons vortices of topological-type, Calc. Var. Partial Differ. Equ., 29 (2007), 191-217.  doi: 10.1007/s00526-006-0062-9.  Google Scholar

[37]

F. Wilczek, Disassembling anyons, Phys. Rev. Lett., 69 (1992), 132-135.  doi: 10.1103/PhysRevLett.69.132.  Google Scholar

[1]

Youngae Lee. Non-topological solutions in a generalized Chern-Simons model on torus. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1315-1330. doi: 10.3934/cpaa.2017064

[2]

Kwangseok Choe, Jongmin Han, Chang-Shou Lin. Bubbling solutions for the Chern-Simons gauged $O(3)$ sigma model in $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2703-2728. doi: 10.3934/dcds.2014.34.2703

[3]

Youngae Lee. Topological solutions in the Maxwell-Chern-Simons model with anomalous magnetic moment. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1293-1314. doi: 10.3934/dcds.2018053

[4]

Yanqin Fang, De Tang. Method of sub-super solutions for fractional elliptic equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3153-3165. doi: 10.3934/dcdsb.2017212

[5]

Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189

[6]

Youyan Wan, Jinggang Tan. The existence of nontrivial solutions to Chern-Simons-Schrödinger systems. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2765-2786. doi: 10.3934/dcds.2017119

[7]

Lingyu Li, Jianfu Yang, Jinge Yang. Solutions to Chern-Simons-Schrödinger systems with external potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1967-1981. doi: 10.3934/dcdss.2021008

[8]

Hartmut Pecher. Local solutions with infinite energy of the Maxwell-Chern-Simons-Higgs system in Lorenz gauge. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2193-2204. doi: 10.3934/dcds.2016.36.2193

[9]

Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete & Continuous Dynamical Systems - S, 2021, 14 (6) : 1931-1944. doi: 10.3934/dcdss.2021016

[10]

Berat Karaagac. New exact solutions for some fractional order differential equations via improved sub-equation method. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 447-454. doi: 10.3934/dcdss.2019029

[11]

Hartmut Pecher. The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4875-4893. doi: 10.3934/dcds.2019199

[12]

Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201

[13]

Hyungjin Huh. Towards the Chern-Simons-Higgs equation with finite energy. Discrete & Continuous Dynamical Systems, 2011, 30 (4) : 1145-1159. doi: 10.3934/dcds.2011.30.1145

[14]

Nikolaos Bournaveas, Timothy Candy, Shuji Machihara. A note on the Chern-Simons-Dirac equations in the Coulomb gauge. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2693-2701. doi: 10.3934/dcds.2014.34.2693

[15]

Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems & Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016

[16]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817

[17]

Songhai Deng, Zhong Wan, Yanjiu Zhou. Optimization model and solution method for dynamically correlated two-product newsvendor problems based on Copula. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1637-1652. doi: 10.3934/dcdss.2020096

[18]

Anthony W. Baker, Michael Dellnitz, Oliver Junge. Topological method for rigorously computing periodic orbits using Fourier modes. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 901-920. doi: 10.3934/dcds.2005.13.901

[19]

Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971

[20]

Daniel Wilczak, Piotr Zgliczyński. Topological method for symmetric periodic orbits for maps with a reversing symmetry. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 629-652. doi: 10.3934/dcds.2007.17.629

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (94)
  • HTML views (112)
  • Cited by (0)

Other articles
by authors

[Back to Top]