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Multi-symplectic method to simulate Soliton resonance of (2+1)-dimensional Boussinesq equation
1. | School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, China, China, China |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
T. Nagasawa and Y. Nishida, Mechanism of resonant interaction of plane ion-acoustic solitons, Phys. Rev. A, 46 (1992), 3471-3476. |
[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. |
[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. |
[10] |
J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, Berlin, 1987. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
W. P. Hu and Z. C. Deng, Multi-symplectic methods for generalized fifth-order KdV equation, Chinese Phys. B, 17 (2008), 3923-3929. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
T. Nagasawa and Y. Nishida, Mechanism of resonant interaction of plane ion-acoustic solitons, Phys. Rev. A, 46 (1992), 3471-3476. |
[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. |
[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. |
[10] |
J. Pedlosky, "Geophysical Fluid Dynamics," Springer-Verlag, Berlin, 1987. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[33] |
W. P. Hu and Z. C. Deng, Multi-symplectic methods for generalized fifth-order KdV equation, Chinese Phys. B, 17 (2008), 3923-3929. |
[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. |
[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. |
[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. |
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