• Previous Article
    Global generalized solvability in the Keller-Segel system with singular sensitivity and arbitrary superlinear degradation
  • DCDS-B Home
  • This Issue
  • Next Article
    Approaching logarithmic singularities in quasilinear chemotaxis-consumption systems with signal-dependent sensitivities
doi: 10.3934/dcdsb.2022003
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.

Finite difference methods for the one-dimensional Chern-Simons gauged models

Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Republic of Korea

*Corresponding author: Bora Moon

Received  August 2021 Revised  November 2021 Early access January 2022

Fund Project: The work of J. Kim and B. Moon was supported by the Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science and ICT (NRF-2020R1A4A3079066) and the work of B. Moon was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2019R1I1A1A01059585)

We present finite difference schemes for the one-dimensional Chern-Simons gauged Schrödinger and Dirac equations. We provide two numerical schemes for the Chern-Simons-Schrödinger equations, each of them has its own advantage in total charge preservation and the second-order accuracy. On the other hand, we offer the second-order, total charge-preserving numerical scheme for the Chern-Simons-Dirac equations. We numerically test each method and validate the total charge preserving properties. We also compare the solutions to the Chern-Simons gauged equations with the equations without the gauge effect, illustrating the effect of gauge fields on the dynamics of the matter field.

Citation: Jeongho Kim, Bora Moon. Finite difference methods for the one-dimensional Chern-Simons gauged models. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022003
References:
[1]

A. Alvarez, Linearized Crank-Nicholson scheme for nonlinear Dirac equations, J. Comput. Phys., 99 (1992), 348-350.  doi: 10.1016/0021-9991(92)90214-J.

[2]

X. AntoineW. Bao and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.  doi: 10.1016/j.cpc.2013.07.012.

[3]

W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 50 (2012), 492-521.  doi: 10.1137/110830800.

[4]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[5]

W. BaoD. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), 318-342.  doi: 10.1016/S0021-9991(03)00102-5.

[6]

W. BaoS. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.  doi: 10.1006/jcph.2001.6956.

[7]

W. BaoS. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.  doi: 10.1137/S1064827501393253.

[8]

W. Bao and X.-G. Li, An efficient and stable numerical method for the Maxwell-Dirac system, J. Comput. Phys., 199 (2004), 663-687.  doi: 10.1016/j.jcp.2004.03.003.

[9]

W. BaoQ. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation, J. Comput. Phys., 235 (2013), 423-445.  doi: 10.1016/j.jcp.2012.10.054.

[10]

L. BergéA. de Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007.

[11]

C. BesseS. DescombesG. Dujardin and I. Lacroix-Violet, Energy-preserving methods for nonlinear Schrödinger equations, IMA J. Numer. Anal., 41 (2021), 618-653.  doi: 10.1093/imanum/drz067.

[12]

N. Bournaveas, A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension, J. Funct. Anal., 173 (2000), 203-213.  doi: 10.1006/jfan.1999.3559.

[13]

N. BournaveasT. Candy and S. Machihara, A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discrete Contin. Dyn. Syst., 34 (2014), 2693-2701.  doi: 10.3934/dcds.2014.34.2693.

[14]

J. ByeonH. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024.

[15]

J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimesion, J. Funct. Anal., 13 (1973), 173-184.  doi: 10.1016/0022-1236(73)90043-8.

[16]

M. Chae and S.-J. Oh, Small data global existence and decay for relativistic Chern-Simons equations, Ann. Henri Poincaré, 18 (2017), 2123-2198.  doi: 10.1007/s00023-016-0547-9.

[17]

Q. ChangE. Jia and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397-415.  doi: 10.1006/jcph.1998.6120.

[18]

S. Demoulini, Global existence for a nonlinear Schroedinger-Chern-Simons system on a surface, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 207-225.  doi: 10.1016/j.anihpc.2006.01.004.

[19]

Q. Du, Discrete gauge invariant approximations of a time dependent Ginzburg-Landau model of superconductivity, Math. Comp., 67 (1998), 965-986.  doi: 10.1090/S0025-5718-98-00954-5.

[20]

G. Dunne, Self-Dual Chern-Simons Theories, Springer, Berlin, 1995.

[21]

F. Fillion-GourdeaE. Lorin and A. D. Bandrauk, Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling, Comput. Phys. Commun., 183 (2012), 1403-1415.  doi: 10.1016/j.cpc.2012.02.012.

[22]

R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., 58 (1992), 83-102.  doi: 10.1090/S0025-5718-1992-1106968-6.

[23]

H. Huh, Cauchy problem for the fermion field equation coupled with the Chern-Simons gauge, Lett. Math. Phys., 79 (2007), 75-94.  doi: 10.1007/s11005-006-0118-y.

[24]

H. Huh, Global solutions and asymptotic behabiors of the Chern-Simons-Dirac equations in $ \mathbb R^{1+1}$, J. Math. Anal. Appl., 366 (2010), 706-713.  doi: 10.1016/j.jmaa.2009.12.055.

[25]

H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., (2013), Article ID 590653, 7 pp. doi: 10.1155/2013/590653.

[26]

H. Huh, Reduction of Chern-Simons-Schrödinger systems in one space dimension, J. Appl. Math., (2013), Article ID 631089, 4 pp. doi: 10.1155/2013/631089.

[27]

H. Huh, Global energy solution of Chern-Simons-Higgs equations in one space dimension, J. Math. Anal. Appl., 420 (2014), 781-791.  doi: 10.1016/j.jmaa.2014.06.013.

[28]

P. A. Horvathy and P. Zhang, Vortices in (abelian) Chern-Simons gauge theory, Phys. Rep., 481 (2009), 83-142.  doi: 10.1016/j.physrep.2009.07.003.

[29]

R. JackiwK. Lee and E. J. Weinberg, Self-dual Chern-Simons solitons, Phys. Rev. D, 42 (1990), 3488-3499.  doi: 10.1103/PhysRevD.42.3488.

[30]

R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500.

[31]

H. C. KaoK. Lee and T. Lee, BPS domain wall solutions in self-dual Chern-Simons-Higgs systems, Phys. Rev. D, 55 (1997), 6447-6453.  doi: 10.1103/PhysRevD.55.6447.

[32]

H. LiZ. Mu and Y. Wang, An energy-preserving Crank-Nicolson Galerkin spectral element method for the two dimensional nonlinear Schrödinger equation, J. Comput. Appl. Math., 344 (2018), 245-258.  doi: 10.1016/j.cam.2018.05.025.

[33]

B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not., (2014), 6341–6398. doi: 10.1093/imrn/rnt161.

[34]

E. LorinS. Chelkowski and A. Bandrauk, A numerical Maxwell-Schrödinger model for intense laser-matter interaction and propagation, Comput. Phys. Commun., 177 (2007), 908-932.  doi: 10.1016/j.cpc.2007.07.005.

[35]

K. Momberger and A. Belkacem, Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heacy-ion collision, Phys. Rev. A, 53 (1996), 1605-1622. 

[36]

C. MüllerN. Grün and W. Scheid, Finite element formulation of the Dirac equation and the problem of fermion doubling, Phys. Lett. A, 242 (1998), 245-250. 

[37]

H. Pecher, The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces, Discrerte Contin. Dyn. Syst., 39 (2019), 4875-4893.  doi: 10.3934/dcds.2019199.

[38]

A. Polychronakos, Abelian Chern-Simons theories in 2+1 dimensions, Ann. Phys., 203 (1990), 231-254.  doi: 10.1016/0003-4916(90)90171-J.

[39]

Y. Tsutsumi, Global existence and uniqueness of energy solutions for the Maxwell-Schrödinger equations in one space dimensions, Hokkaido Math. J., 24 (1995), 617-639.  doi: 10.14492/hokmj/1380892611.

[40]

J. TworzydloC. W. Groth and C. W. J. Beenakker, Finite difference method for transport properties of massless Dirac fermions, Phys. Rev. B, 78 (2008), 235438.  doi: 10.1103/PhysRevB.78.235438.

[41]

J. XuS. Shao and H. Tang, Numerical methods for non-linear Dirac equation, J. Comput. Phys., 245 (2013), 131-149.  doi: 10.1016/j.jcp.2013.03.031.

show all references

References:
[1]

A. Alvarez, Linearized Crank-Nicholson scheme for nonlinear Dirac equations, J. Comput. Phys., 99 (1992), 348-350.  doi: 10.1016/0021-9991(92)90214-J.

[2]

X. AntoineW. Bao and C. Besse, Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations, Comput. Phys. Commun., 184 (2013), 2621-2633.  doi: 10.1016/j.cpc.2013.07.012.

[3]

W. Bao and Y. Cai, Uniform error estimates of finite difference methods for the nonlinear Schrödinger equation with wave operator, SIAM J. Numer. Anal., 50 (2012), 492-521.  doi: 10.1137/110830800.

[4]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Models, 6 (2013), 1-135.  doi: 10.3934/krm.2013.6.1.

[5]

W. BaoD. Jaksch and P. A. Markowich, Numerical solution of the Gross-Pitaevskii equation for Bose-Einstein condensation, J. Comput. Phys., 187 (2003), 318-342.  doi: 10.1016/S0021-9991(03)00102-5.

[6]

W. BaoS. Jin and P. A. Markowich, On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime, J. Comput. Phys., 175 (2002), 487-524.  doi: 10.1006/jcph.2001.6956.

[7]

W. BaoS. Jin and P. A. Markowich, Numerical study of time-splitting spectral discretizations of nonlinear Schrödinger equations in the semiclassical regimes, SIAM J. Sci. Comput., 25 (2003), 27-64.  doi: 10.1137/S1064827501393253.

[8]

W. Bao and X.-G. Li, An efficient and stable numerical method for the Maxwell-Dirac system, J. Comput. Phys., 199 (2004), 663-687.  doi: 10.1016/j.jcp.2004.03.003.

[9]

W. BaoQ. Tang and Z. Xu, Numerical methods and comparison for computing dark and bright solitons in the nonlinear Schrödinger equation, J. Comput. Phys., 235 (2013), 423-445.  doi: 10.1016/j.jcp.2012.10.054.

[10]

L. BergéA. de Bouard and J.-C. Saut, Blowing up time-dependent solutions of the planar, Chern-Simons gauged nonlinear Schrödinger equation, Nonlinearity, 8 (1995), 235-253.  doi: 10.1088/0951-7715/8/2/007.

[11]

C. BesseS. DescombesG. Dujardin and I. Lacroix-Violet, Energy-preserving methods for nonlinear Schrödinger equations, IMA J. Numer. Anal., 41 (2021), 618-653.  doi: 10.1093/imanum/drz067.

[12]

N. Bournaveas, A new proof of global existence for the Dirac Klein-Gordon equations in one space dimension, J. Funct. Anal., 173 (2000), 203-213.  doi: 10.1006/jfan.1999.3559.

[13]

N. BournaveasT. Candy and S. Machihara, A note on the Chern-Simons-Dirac equations in the Coulomb gauge, Discrete Contin. Dyn. Syst., 34 (2014), 2693-2701.  doi: 10.3934/dcds.2014.34.2693.

[14]

J. ByeonH. Huh and J. Seok, Standing waves of nonlinear Schrödinger equations with the gauge field, J. Funct. Anal., 263 (2012), 1575-1608.  doi: 10.1016/j.jfa.2012.05.024.

[15]

J. M. Chadam, Global solutions of the Cauchy problem for the (classical) coupled Maxwell-Dirac equations in one space dimesion, J. Funct. Anal., 13 (1973), 173-184.  doi: 10.1016/0022-1236(73)90043-8.

[16]

M. Chae and S.-J. Oh, Small data global existence and decay for relativistic Chern-Simons equations, Ann. Henri Poincaré, 18 (2017), 2123-2198.  doi: 10.1007/s00023-016-0547-9.

[17]

Q. ChangE. Jia and W. Sun, Difference schemes for solving the generalized nonlinear Schrödinger equation, J. Comput. Phys., 148 (1999), 397-415.  doi: 10.1006/jcph.1998.6120.

[18]

S. Demoulini, Global existence for a nonlinear Schroedinger-Chern-Simons system on a surface, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 207-225.  doi: 10.1016/j.anihpc.2006.01.004.

[19]

Q. Du, Discrete gauge invariant approximations of a time dependent Ginzburg-Landau model of superconductivity, Math. Comp., 67 (1998), 965-986.  doi: 10.1090/S0025-5718-98-00954-5.

[20]

G. Dunne, Self-Dual Chern-Simons Theories, Springer, Berlin, 1995.

[21]

F. Fillion-GourdeaE. Lorin and A. D. Bandrauk, Numerical solution of the time-dependent Dirac equation in coordinate space without fermion-doubling, Comput. Phys. Commun., 183 (2012), 1403-1415.  doi: 10.1016/j.cpc.2012.02.012.

[22]

R. T. Glassey, Convergence of an energy-preserving scheme for the Zakharov equations in one space dimension, Math. Comp., 58 (1992), 83-102.  doi: 10.1090/S0025-5718-1992-1106968-6.

[23]

H. Huh, Cauchy problem for the fermion field equation coupled with the Chern-Simons gauge, Lett. Math. Phys., 79 (2007), 75-94.  doi: 10.1007/s11005-006-0118-y.

[24]

H. Huh, Global solutions and asymptotic behabiors of the Chern-Simons-Dirac equations in $ \mathbb R^{1+1}$, J. Math. Anal. Appl., 366 (2010), 706-713.  doi: 10.1016/j.jmaa.2009.12.055.

[25]

H. Huh, Energy solution to the Chern-Simons-Schrödinger equations, Abstr. Appl. Anal., (2013), Article ID 590653, 7 pp. doi: 10.1155/2013/590653.

[26]

H. Huh, Reduction of Chern-Simons-Schrödinger systems in one space dimension, J. Appl. Math., (2013), Article ID 631089, 4 pp. doi: 10.1155/2013/631089.

[27]

H. Huh, Global energy solution of Chern-Simons-Higgs equations in one space dimension, J. Math. Anal. Appl., 420 (2014), 781-791.  doi: 10.1016/j.jmaa.2014.06.013.

[28]

P. A. Horvathy and P. Zhang, Vortices in (abelian) Chern-Simons gauge theory, Phys. Rep., 481 (2009), 83-142.  doi: 10.1016/j.physrep.2009.07.003.

[29]

R. JackiwK. Lee and E. J. Weinberg, Self-dual Chern-Simons solitons, Phys. Rev. D, 42 (1990), 3488-3499.  doi: 10.1103/PhysRevD.42.3488.

[30]

R. Jackiw and S.-Y. Pi, Classical and quantal nonrelativistic Chern-Simons theory, Phys. Rev. D, 42 (1990), 3500-3513.  doi: 10.1103/PhysRevD.42.3500.

[31]

H. C. KaoK. Lee and T. Lee, BPS domain wall solutions in self-dual Chern-Simons-Higgs systems, Phys. Rev. D, 55 (1997), 6447-6453.  doi: 10.1103/PhysRevD.55.6447.

[32]

H. LiZ. Mu and Y. Wang, An energy-preserving Crank-Nicolson Galerkin spectral element method for the two dimensional nonlinear Schrödinger equation, J. Comput. Appl. Math., 344 (2018), 245-258.  doi: 10.1016/j.cam.2018.05.025.

[33]

B. Liu, P. Smith and D. Tataru, Local wellposedness of Chern-Simons-Schrödinger, Int. Math. Res. Not., (2014), 6341–6398. doi: 10.1093/imrn/rnt161.

[34]

E. LorinS. Chelkowski and A. Bandrauk, A numerical Maxwell-Schrödinger model for intense laser-matter interaction and propagation, Comput. Phys. Commun., 177 (2007), 908-932.  doi: 10.1016/j.cpc.2007.07.005.

[35]

K. Momberger and A. Belkacem, Numerical treatment of the time-dependent Dirac equation in momentum space for atomic processes in relativistic heacy-ion collision, Phys. Rev. A, 53 (1996), 1605-1622. 

[36]

C. MüllerN. Grün and W. Scheid, Finite element formulation of the Dirac equation and the problem of fermion doubling, Phys. Lett. A, 242 (1998), 245-250. 

[37]

H. Pecher, The Chern-Simons-Higgs and the Chern-Simons-Dirac equations in Fourier-Lebesgue spaces, Discrerte Contin. Dyn. Syst., 39 (2019), 4875-4893.  doi: 10.3934/dcds.2019199.

[38]

A. Polychronakos, Abelian Chern-Simons theories in 2+1 dimensions, Ann. Phys., 203 (1990), 231-254.  doi: 10.1016/0003-4916(90)90171-J.

[39]

Y. Tsutsumi, Global existence and uniqueness of energy solutions for the Maxwell-Schrödinger equations in one space dimensions, Hokkaido Math. J., 24 (1995), 617-639.  doi: 10.14492/hokmj/1380892611.

[40]

J. TworzydloC. W. Groth and C. W. J. Beenakker, Finite difference method for transport properties of massless Dirac fermions, Phys. Rev. B, 78 (2008), 235438.  doi: 10.1103/PhysRevB.78.235438.

[41]

J. XuS. Shao and H. Tang, Numerical methods for non-linear Dirac equation, J. Comput. Phys., 245 (2013), 131-149.  doi: 10.1016/j.jcp.2013.03.031.

Figure 1.  Numerical results for first-order scheme
Figure 2.  Numerical results for second-order scheme
Figure 3.  Dynamics of the discrete total charge for each scheme
Figure 4.  Comparison between the solutions of first-and second order schemes
Figure 5.  Comparison between the solutions to the CSS equations and the Schrödinger equation
Figure 6.  Numerical results of the numerical scheme for CSD equations
Figure 7.  Dynamics of the discrete total charge for each scheme
Figure 8.  Comparison between the solutions to the CSD equations and the Dirac equations at $ t = 5 $
Figure 9.  Dynamics of the discrete total energy for each scheme
[1]

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

[2]

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

[3]

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

[4]

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

[5]

Jianjun Yuan. Global existence and scattering of equivariant defocusing Chern-Simons-Schrödinger system. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5541-5570. doi: 10.3934/dcds.2020237

[6]

Jincai Kang, Chunlei Tang. Existence of nontrivial solutions to Chern-Simons-Schrödinger system with indefinite potential. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021016

[7]

Sigmund Selberg, Achenef Tesfahun. Global well-posedness of the Chern-Simons-Higgs equations with finite energy. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2531-2546. doi: 10.3934/dcds.2013.33.2531

[8]

Jeongho Kim, Bora Moon. Hydrodynamic limits of the nonlinear Schrödinger equation with the Chern-Simons gauge fields. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2541-2561. doi: 10.3934/dcds.2021202

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Jianjun Yuan. On the well-posedness of Maxwell-Chern-Simons-Higgs system in the Lorenz gauge. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2389-2403. doi: 10.3934/dcds.2014.34.2389

[15]

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

[16]

Hsin-Yuan Huang. Vortex Condensation in General U(1)×U(1) Abelian Chern-Simons Model on a flat torus. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4415-4428. doi: 10.3934/dcdsb.2021234

[17]

Kei Nakamura, Tohru Ozawa. Finite charge solutions to cubic Schrödinger equations with a nonlocal nonlinearity in one space dimension. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 789-801. doi: 10.3934/dcds.2013.33.789

[18]

Zhi-You Chen, Chung-Yang Wang, Yu-Jen Huang. On the asymptotic behavior of solutions for the self-dual Maxwell-Chern-Simons $ O(3) $ Sigma model. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022077

[19]

Katherine A. Kime. Palindromic control and mirror symmetries in finite difference discretizations of 1-D Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (4) : 1601-1621. doi: 10.3934/dcdsb.2018063

[20]

Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic and Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (240)
  • HTML views (141)
  • Cited by (0)

Other articles
by authors

[Back to Top]