American Institute of Mathematical Sciences

Advanced
September  2013, 5(3): 295-318. doi: 10.3934/jgm.2013.5.295

Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation

Weipeng Hu 1, , Zichen Deng 1, and  Yuyue Qin 1,

1. 

School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, China, China, China

Received  January 2013 Revised  July 2013 Published  September 2013

The soliton interactions, especially the soliton resonance phenomena of the (2+1)-dimensional Boussinesq equation have been investigated numerically in this paper. Based on the Bridges's multi-symplectic idea, the multi-symplectic formulations with several conservation laws for the (2+1)-dimensional Boussinesq equation are presented firstly. Then, a forty-five points implicit multi-symplectic scheme is constructed. Finally, according to the soliton resonance condition, numerical experiments on the two-soliton solution of the (2+1)-dimensional Boussinesq equation for simulating the soliton interaction phenomena, especially the soliton resonance are reported. From the results of the numerical experiments, it can be concluded that the multi-symplectic scheme can simulate the soliton resonance phenomena perfectly, which can be used to make further investigation on the interaction and the energy distribution of gravity waves, and evaluate the impact on the ship traffic on the surface of water.
Keywords: conservation law., (2+1)-dimensional Boussinesq Equation, Soliton interaction, soliton resonance, multi-symplectic method.
Mathematics Subject Classification: Primary: 35G20, 65M06; Secondary: 35Q3.
Citation: Weipeng Hu, Zichen Deng, Yuyue Qin. Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation. Journal of Geometric Mechanics, 2013, 5 (3) : 295-318. doi: 10.3934/jgm.2013.5.295
References:
[1]

F. Kako and N. Yajima, Interaction of ion-acoustic solitons in two-dimensional space, Journal of the Physical Society of Japan, 49 (1980), 2063-2071. doi: 10.1143/JPSJ.49.2063.  Google Scholar

[2]

M. Tajiri and H. Maesono, Resonant interactions of drift vortex solitons in a convective motion of a plasma, Physical Review E, 55 (1997), 3351-3357. doi: 10.1103/PhysRevE.55.3351.  Google Scholar

[3]

Y. Nakamura, H. Bailung and K. E. Lonngren, Oblique collision of modified Korteweg-de Vries ion-acoustic solitons, Physics of Plasmas, 6 (1999), 3466-3470. doi: 10.1063/1.873607.  Google Scholar

[4]

K. I. Maruno and G. Biondini, Resonance and web structure in discrete soliton systems: the two-dimensional Toda Lattice and its fully discrete and ultra-discrete analogues, Journal of Physics A: Mathematical and General, 37 (2004), 11819-11839. doi: 10.1088/0305-4470/37/49/005.  Google Scholar

[5]

P. A. Folkes, H. Ikezi and R. Davis, Two-dimensional interaction of ion-acoustic solitons, Physical Review Letters, 45 (1980), 902-904. doi: 10.1103/PhysRevLett.45.902.  Google Scholar

[6]

Y. Nishida and T. Nagasawa, Oblique collision of plane ion-acoustic solitons, Physical Review Letters, 45 (1980), 1626-1629. doi: 10.1103/PhysRevLett.45.1626.  Google Scholar

[7]

T. Nagasawa and Y. Nishida, Mechanism of resonant interaction of plane ion-acoustic solitons, Phys. Rev. A, 46 (1992), 3471-3476. Google Scholar

[8]

A. R. Osborne, M. Onorato, M. Serio and L. Bergamasco, Soliton creation and destruction, resonant interactions, and inelastic collisions in shallow water waves, Physical Review Letters, 81 (1998), 3559-3562. doi: 10.1103/PhysRevLett.81.3559.  Google Scholar

[9]

J. Sreekumar and V. M. Nandakumaran, Soliton resonances in Helium films, Physics Letters A, 112 (1985), 168-170. doi: 10.1016/0375-9601(85)90681-4.  Google Scholar

[10]

J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, Berlin, 1987. Google Scholar

[11]

R. Ibragimov, Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 593-623. doi: 10.1016/j.cnsns.2006.06.011.  Google Scholar

[12]

T. Soomere, Fast ferry traffic as a qualitatively new forcing factor of environmental processes in non-tidal sea areas: A case study in Tallinn bay, Baltic Sea, Environmental Fluid Mechanics, 5 (2005), 293-323. doi: 10.1007/s10652-005-5226-1.  Google Scholar

[13]

T. Soomere and J. Engelbrecht, Weakly two-dimensional interaction of solitons in shallow water, European Journal of Mechanics - B/Fluids, 25 (2006), 636-648. doi: 10.1016/j.euromechflu.2006.02.008.  Google Scholar

[14]

R. Hirota, Exact envelope-soliton solutions of a nonlinear wave-equation, Journal of Mathematical Physics, 14 (1973), 805-809. doi: 10.1063/1.1666399.  Google Scholar

[15]

R. Hirota, Exact N-soliton solutions of wave-equation of long waves in shallow-water and in nonlinear lattices, Journal of Mathematical Physics, 14 (1973), 810-814. doi: 10.1063/1.1666400.  Google Scholar

[16]

J. J. C. Nimmo and N. C. Freeman, A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a Wronskian, Physics Letters A, 95 (1983), 4-6. doi: 10.1016/0375-9601(83)90765-X.  Google Scholar

[17]

O. V. Kaptsov, Construction of exact solutions of the Boussinesq equation, Journal of Applied Mechanics and Technical Physics, 39 (1998), 389-392. doi: 10.1007/BF02468120.  Google Scholar

[18]

A. M. Wazwaz, Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, Solitons & Fractals, 12 (2001), 1549-1556. doi: 10.1016/S0960-0779(00)00133-8.  Google Scholar

[19]

Z. Y. Yan and G. Bluman, New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations, Computer Physics Communications, 149 (2002), 11-18. doi: 10.1016/S0010-4655(02)00587-8.  Google Scholar

[20]

Y. Zhang and D. Y. Chen, A modified Bäcklund transformation and multi-soliton solution for the Boussinesq equation, Chaos, Solitons & Fractals, 23 (2005), 175-181. doi: 10.1016/j.chaos.2004.04.006.  Google Scholar

[21]

A. M. Wazwaz, Multiple-soliton solutions for the Boussinesq equation, Applied Mathematics and Computation, 192 (2007), 479-486. doi: 10.1016/j.amc.2007.03.023.  Google Scholar

[22]

W. P. Zeng, L. Y. Huang and M. Z. Qin, The multi-symplectic algorithm for "Good" Boussinesq equation, Applied Mathematics and Mechanics, 23 (2002), 835-841. doi: 10.1007/BF02456980.  Google Scholar

[23]

H. El-Zoheiry, Numerical investigation for the solitary waves interaction of the "Good" Boussinesq equation, Applied Numerical Mathematics, 45 (2003), 161-173. doi: 10.1016/S0168-9274(02)00187-3.  Google Scholar

[24]

W. P. Hu and Z. C. Deng, Multi-symplectic method for generalized Boussinesq equation, Applied Mathematics and Mechanics, 29 (2008), 927-932. doi: 10.1007/s10483-008-0711-3.  Google Scholar

[25]

K. B. Blyuss, T. J. Bridges and G. Derks, Transverse instability and its long-term development for solitary waves of the (2+1)-dimensional Boussinesq equation, Physical Review E, 67 (2003), 056626. doi: 10.1103/PhysRevE.67.056626.  Google Scholar

[26]

Y. Chen, Z. Y. Yan and H. Zhang, New explicit solitary wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation, Physics Letters A, 307 (2003), 107-113. doi: 10.1016/S0375-9601(02)01668-7.  Google Scholar

[27]

H. Q. Zhang, X. H. Meng, J. Li and B. Tian, Soliton resonance of the (2+1)-dimensional Boussinesq equation for gravity water waves, Nonlinear Analysis: Real World Applications, 9 (2008), 920-926. doi: 10.1016/j.nonrwa.2007.01.010.  Google Scholar

[28]

T. J. Bridges, Multi-symplectic structures and wave propagation, Mathematical Proceedings of the Cambridge Philosophical Society, 121 (1997), 147-190. doi: 10.1017/S0305004196001429.  Google Scholar

[29]

S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, Journal of Computational Physics, 157 (2000), 473-499. doi: 10.1006/jcph.1999.6372.  Google Scholar

[30]

T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Physics Letters A, 284 (2001), 184-193. doi: 10.1016/S0375-9601(01)00294-8.  Google Scholar

[31]

A. L. Islas and C. M. Schober, Multi-symplectic methods for generalized Schrödinger equations, Future Generation Computer Systems, 19 (2003), 403-413. doi: 10.1016/S0167-739X(02)00167-X.  Google Scholar

[32]

B. E. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs, Future Generation Computer Systems, 19 (2003), 395-402. doi: 10.1016/S0167-739X(02)00166-8.  Google Scholar

[33]

W. P. Hu and Z. C. Deng, Multi-symplectic methods for generalized fifth-order KdV equation, Chinese Phys. B, 17 (2008), 3923-3929. Google Scholar

[34]

W. P. Hu and Z. C. Deng, Multi-symplectic methods to analyze the mixed state of II-superconductors, Science in China Series G: Physics, Mechanics and Astronomy, 51 (2008), 1835-1844. doi: 10.1007/s11433-008-0192-5.  Google Scholar

[35]

R. S. Johnson, A two-dimensional Boussinesq equation for water waves and some of its solutions, Journal of Fluid Mechanics, 323 (1996), 65-78. doi: 10.1017/S0022112096000845.  Google Scholar

[36]

T. J. Bridges and S. Reich, Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Physica D: Nonlinear Phenomena, 152-153 (2001), 491-504. doi: 10.1016/S0167-2789(01)00188-9.  Google Scholar

show all references

References:
[1]

F. Kako and N. Yajima, Interaction of ion-acoustic solitons in two-dimensional space, Journal of the Physical Society of Japan, 49 (1980), 2063-2071. doi: 10.1143/JPSJ.49.2063.  Google Scholar

[2]

M. Tajiri and H. Maesono, Resonant interactions of drift vortex solitons in a convective motion of a plasma, Physical Review E, 55 (1997), 3351-3357. doi: 10.1103/PhysRevE.55.3351.  Google Scholar

[3]

Y. Nakamura, H. Bailung and K. E. Lonngren, Oblique collision of modified Korteweg-de Vries ion-acoustic solitons, Physics of Plasmas, 6 (1999), 3466-3470. doi: 10.1063/1.873607.  Google Scholar

[4]

K. I. Maruno and G. Biondini, Resonance and web structure in discrete soliton systems: the two-dimensional Toda Lattice and its fully discrete and ultra-discrete analogues, Journal of Physics A: Mathematical and General, 37 (2004), 11819-11839. doi: 10.1088/0305-4470/37/49/005.  Google Scholar

[5]

P. A. Folkes, H. Ikezi and R. Davis, Two-dimensional interaction of ion-acoustic solitons, Physical Review Letters, 45 (1980), 902-904. doi: 10.1103/PhysRevLett.45.902.  Google Scholar

[6]

Y. Nishida and T. Nagasawa, Oblique collision of plane ion-acoustic solitons, Physical Review Letters, 45 (1980), 1626-1629. doi: 10.1103/PhysRevLett.45.1626.  Google Scholar

[7]

T. Nagasawa and Y. Nishida, Mechanism of resonant interaction of plane ion-acoustic solitons, Phys. Rev. A, 46 (1992), 3471-3476. Google Scholar

[8]

A. R. Osborne, M. Onorato, M. Serio and L. Bergamasco, Soliton creation and destruction, resonant interactions, and inelastic collisions in shallow water waves, Physical Review Letters, 81 (1998), 3559-3562. doi: 10.1103/PhysRevLett.81.3559.  Google Scholar

[9]

J. Sreekumar and V. M. Nandakumaran, Soliton resonances in Helium films, Physics Letters A, 112 (1985), 168-170. doi: 10.1016/0375-9601(85)90681-4.  Google Scholar

[10]

J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, Berlin, 1987. Google Scholar

[11]

R. Ibragimov, Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves, Communications in Nonlinear Science and Numerical Simulation, 13 (2008), 593-623. doi: 10.1016/j.cnsns.2006.06.011.  Google Scholar

[12]

T. Soomere, Fast ferry traffic as a qualitatively new forcing factor of environmental processes in non-tidal sea areas: A case study in Tallinn bay, Baltic Sea, Environmental Fluid Mechanics, 5 (2005), 293-323. doi: 10.1007/s10652-005-5226-1.  Google Scholar

[13]

T. Soomere and J. Engelbrecht, Weakly two-dimensional interaction of solitons in shallow water, European Journal of Mechanics - B/Fluids, 25 (2006), 636-648. doi: 10.1016/j.euromechflu.2006.02.008.  Google Scholar

[14]

R. Hirota, Exact envelope-soliton solutions of a nonlinear wave-equation, Journal of Mathematical Physics, 14 (1973), 805-809. doi: 10.1063/1.1666399.  Google Scholar

[15]

R. Hirota, Exact N-soliton solutions of wave-equation of long waves in shallow-water and in nonlinear lattices, Journal of Mathematical Physics, 14 (1973), 810-814. doi: 10.1063/1.1666400.  Google Scholar

[16]

J. J. C. Nimmo and N. C. Freeman, A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a Wronskian, Physics Letters A, 95 (1983), 4-6. doi: 10.1016/0375-9601(83)90765-X.  Google Scholar

[17]

O. V. Kaptsov, Construction of exact solutions of the Boussinesq equation, Journal of Applied Mechanics and Technical Physics, 39 (1998), 389-392. doi: 10.1007/BF02468120.  Google Scholar

[18]

A. M. Wazwaz, Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method, Chaos, Solitons & Fractals, 12 (2001), 1549-1556. doi: 10.1016/S0960-0779(00)00133-8.  Google Scholar

[19]

Z. Y. Yan and G. Bluman, New compacton soliton solutions and solitary patterns solutions of nonlinearly dispersive Boussinesq equations, Computer Physics Communications, 149 (2002), 11-18. doi: 10.1016/S0010-4655(02)00587-8.  Google Scholar

[20]

Y. Zhang and D. Y. Chen, A modified Bäcklund transformation and multi-soliton solution for the Boussinesq equation, Chaos, Solitons & Fractals, 23 (2005), 175-181. doi: 10.1016/j.chaos.2004.04.006.  Google Scholar

[21]

A. M. Wazwaz, Multiple-soliton solutions for the Boussinesq equation, Applied Mathematics and Computation, 192 (2007), 479-486. doi: 10.1016/j.amc.2007.03.023.  Google Scholar

[22]

W. P. Zeng, L. Y. Huang and M. Z. Qin, The multi-symplectic algorithm for "Good" Boussinesq equation, Applied Mathematics and Mechanics, 23 (2002), 835-841. doi: 10.1007/BF02456980.  Google Scholar

[23]

H. El-Zoheiry, Numerical investigation for the solitary waves interaction of the "Good" Boussinesq equation, Applied Numerical Mathematics, 45 (2003), 161-173. doi: 10.1016/S0168-9274(02)00187-3.  Google Scholar

[24]

W. P. Hu and Z. C. Deng, Multi-symplectic method for generalized Boussinesq equation, Applied Mathematics and Mechanics, 29 (2008), 927-932. doi: 10.1007/s10483-008-0711-3.  Google Scholar

[25]

K. B. Blyuss, T. J. Bridges and G. Derks, Transverse instability and its long-term development for solitary waves of the (2+1)-dimensional Boussinesq equation, Physical Review E, 67 (2003), 056626. doi: 10.1103/PhysRevE.67.056626.  Google Scholar

[26]

Y. Chen, Z. Y. Yan and H. Zhang, New explicit solitary wave solutions for (2+1)-dimensional Boussinesq equation and (3+1)-dimensional KP equation, Physics Letters A, 307 (2003), 107-113. doi: 10.1016/S0375-9601(02)01668-7.  Google Scholar

[27]

H. Q. Zhang, X. H. Meng, J. Li and B. Tian, Soliton resonance of the (2+1)-dimensional Boussinesq equation for gravity water waves, Nonlinear Analysis: Real World Applications, 9 (2008), 920-926. doi: 10.1016/j.nonrwa.2007.01.010.  Google Scholar

[28]

T. J. Bridges, Multi-symplectic structures and wave propagation, Mathematical Proceedings of the Cambridge Philosophical Society, 121 (1997), 147-190. doi: 10.1017/S0305004196001429.  Google Scholar

[29]

S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, Journal of Computational Physics, 157 (2000), 473-499. doi: 10.1006/jcph.1999.6372.  Google Scholar

[30]

T. J. Bridges and S. Reich, Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Physics Letters A, 284 (2001), 184-193. doi: 10.1016/S0375-9601(01)00294-8.  Google Scholar

[31]

A. L. Islas and C. M. Schober, Multi-symplectic methods for generalized Schrödinger equations, Future Generation Computer Systems, 19 (2003), 403-413. doi: 10.1016/S0167-739X(02)00167-X.  Google Scholar

[32]

B. E. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs, Future Generation Computer Systems, 19 (2003), 395-402. doi: 10.1016/S0167-739X(02)00166-8.  Google Scholar

[33]

W. P. Hu and Z. C. Deng, Multi-symplectic methods for generalized fifth-order KdV equation, Chinese Phys. B, 17 (2008), 3923-3929. Google Scholar

[34]

W. P. Hu and Z. C. Deng, Multi-symplectic methods to analyze the mixed state of II-superconductors, Science in China Series G: Physics, Mechanics and Astronomy, 51 (2008), 1835-1844. doi: 10.1007/s11433-008-0192-5.  Google Scholar

[35]

R. S. Johnson, A two-dimensional Boussinesq equation for water waves and some of its solutions, Journal of Fluid Mechanics, 323 (1996), 65-78. doi: 10.1017/S0022112096000845.  Google Scholar

[36]

T. J. Bridges and S. Reich, Multi-symplectic spectral discretizations for the Zakharov-Kuznetsov and shallow water equations, Physica D: Nonlinear Phenomena, 152-153 (2001), 491-504. doi: 10.1016/S0167-2789(01)00188-9.  Google Scholar

[1]

Justin Holmer, Maciej Zworski. Slow soliton interaction with delta impurities. Journal of Modern Dynamics, 2007, 1 (4) : 689-718. doi: 10.3934/jmd.2007.1.689
[2]

Yufeng Zhang, Wen-Xiu Ma, Jin-Yun Yang. A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2941-2948. doi: 10.3934/dcdss.2020167
[3]

John P. Albert. A uniqueness result for 2-soliton solutions of the Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3635-3670. doi: 10.3934/dcds.2019149
[4]

Stephen C. Anco, Maria Luz Gandarias, Elena Recio. Conservation laws and line soliton solutions of a family of modified KP equations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2655-2665. doi: 10.3934/dcdss.2020225
[5]

Ben Muatjetjeja, Dimpho Millicent Mothibi, Chaudry Masood Khalique. Lie group classification a generalized coupled (2+1)-dimensional hyperbolic system. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2803-2812. doi: 10.3934/dcdss.2020219
[6]

Hartmut Pecher. Low regularity solutions for the (2+1)-dimensional Maxwell-Klein-Gordon equations in temporal gauge. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2203-2219. doi: 10.3934/cpaa.2016034
[7]

Liren Lin, Tai-Peng Tsai. Mixed dimensional infinite soliton trains for nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 295-336. doi: 10.3934/dcds.2017013
[8]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309
[9]

Hristo Genev, George Venkov. Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 903-923. doi: 10.3934/dcdss.2012.5.903
[10]

Georgi Grahovski, Rossen Ivanov. Generalised Fourier transform and perturbations to soliton equations. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 579-595. doi: 10.3934/dcdsb.2009.12.579
[11]

Martin Bauer, Martins Bruveris, Philipp Harms, Peter W. Michor. Soliton solutions for the elastic metric on spaces of curves. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1161-1185. doi: 10.3934/dcds.2018049
[12]

Agust Sverrir Egilsson. On embedding the $1:1:2$ resonance space in a Poisson manifold. Electronic Research Announcements, 1995, 1: 48-56.

[13]

Xiangsheng Xu. Global existence of strong solutions to a biological network formulation model in 2+1 dimensions. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6289-6307. doi: 10.3934/dcds.2020280
[14]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098
[15]

Reza Mazrooei-Sebdani, Zahra Yousefi. The coupled 1:2 resonance in a symmetric case and parametric amplification model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3737-3765. doi: 10.3934/dcdsb.2020255
[16]

Afaf Bouharguane. On the instability of a nonlocal conservation law. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 419-426. doi: 10.3934/dcdss.2012.5.419
[17]

Arno Berger. Multi-dimensional dynamical systems and Benford's Law. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 219-237. doi: 10.3934/dcds.2005.13.219
[18]

A. Alexandrou Himonas, Gerson Petronilho. A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2031-2050. doi: 10.3934/dcds.2020351
[19]

W. Josh Sonnier, C. I. Christov. Repelling soliton collisions in coupled Schrödinger equations with negative cross modulation. Conference Publications, 2009, 2009 (Special) : 708-718. doi: 10.3934/proc.2009.2009.708
[20]

Hung Luong. Local well-posedness for the Zakharov system on the background of a line soliton. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2657-2682. doi: 10.3934/cpaa.2018126

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences