# American Institute of Mathematical Sciences

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

 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.
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. [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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##### 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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