In the paper, the coupled 1D Klein-Gordon-Zakharov system (KGZ-equations in short) is considered as the model equation for wave-wave interaction in ionic media. A finite difference scheme is derived for the model equations. A new six point scheme, which is equivalent to the multi-symplectic integrator, is derived. The numerical simulation is also presented for the model equations. Keywords: Coupled 1D Klein-Gordon-Zakharov system; Energy conservation; Six-point schem

Kumar, Arun

Pankaj, R. D.

Sachdeva, Jyoti

English

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Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   53  A New Six Point Finite Difference Scheme for Nonlinear Waves Interaction Model  Dr. Arun Kumar* Department of Mathematics, Government College, Kota (Raj.), India, Dr. R. D. Pankaj Department of Mathematics, J.N.V. University, Jodhpur (Raj.), India Dr. Jyoti Sachdeva Department of Mathematics, M.V.N. University, Palwal (Haryana), India   Abstract: In the paper, the coupled 1D Klein-Gordon-Zakharov system (KGZ-equations in short) is considered as the model equation for wave-wave interaction in ionic media. A finite difference scheme is derived for the model equations. A new six point scheme, which is equivalent to the multi-symplectic integrator, is derived. The numerical simulation is also presented for the model equations. Keywords: Coupled 1D Klein-Gordon-Zakharov system; Energy conservation; Six-point scheme  1. Introduction  Nonlinear evolution equations have a major role in various scientific fields, such as fluid mechanics, plasma physics, optical fibres, solid state physics, chemical physics, and geochemistry. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In recent years, solving nonlinear evolution equations has become a valuable task in many scientific areas including applied mathematics as well as the physical sciences and engineering. For this purpose, some accurate methods have been presented, such as Inverse scattering transform method[1], Bäcklund transformation method[2], Jacobi elliptic function method[3], F-expansion method[4], Hirota’s bilinear method[5], the extended hyperbolic functions method[6, 7], Homotopy perturbation method[8], Bifurcation method [9–11] and so on. Wave-wave interaction is an important problem for both physical and mathematical reasons. Physically, the wave-wave interaction or the wave collisions are common phenomena in science and engineering for both solitary and non-solitary waves. Mathematically solitary wave collision is a major branch of nonlinear wave interaction in ionic media. We consider as the model equation the following Klein-Gordon-Zakharov (KGZ) equations       0,02xxxxttxxttEnnnEEEE                      (1) This system is describe [12, 13] the interaction of a Langmuir wave and an ion sound wave in plasma. More precisely, E is the fast scale component of the electric field, where η denotes the deviation of ion density, in the paper, we discretize the system with finite difference schemes to show the multi-symplectic structure of CNLKGZ system.  Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   54  2. A difference scheme for CNLKGZ system We consider the following generalized CNLKGZ system 0,02xxxxttxxttEnnnEEEE Where           , , , , ,E p x t iq x y x t i x y               (2) We have       (3)                                      Introducing the canonical momenta     2 2, , , , , , , , ,x x t t x x t t x xp a q b p c q d e f g h u p v q            The above system can be written in the following form   zSLzKz zxt                                                                                       (4) with independent variable (t, x)R2 and state variable zRd, d ≥ 2. Here K, LRdxd are two skew-symmetric matrices and S : Rd →R is a scalar-valued smooth function.  z is the standard  gradient in Rd. For S(z) and  z S(z) , the system is multi-symplectic in the sense that K is a skew-symmetric matrix representative of the t direction and L is a skew-symmetric matrix representative of the x direction. S represents a Hamiltonian function [14, 15, 16]. The equation (4) is multi-symplectic in nature with the state variables   1622,,,,,,,,,,,,,,, Rqpvuhgfedcbaqpz T     So the    Tz qpvuhgfedcbakqkpzS 22,,,,,,,,,,,,,,,        1 ; 1kp p q kq p q          We get          2 21 1, ,0 0, , , ,,x t x tx x t tt x x x t xx x t tx xa c p q b d p qp a q b p c q dg e u v h fe f g hu p v q                          and the pair of skew symmetric matrix K and L are          2 21 10 0xx tt xx tttt xx tt xxxx xxp p p q q q p qp q                  Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   55  0000000000000000000000000000000000000000000000000000000000000000000000001000000000000000010000000000000000000000000000000000000000001000000000000000010000000000000000000000001000000000000000010000000000000000000000000000000000000000001000000000000000010000K  and  1000000000000000010000000000000000000000000000000000000000000000000000000000000000000000000000000000000010000000000000000100000000000010000000000011000100000000000000000000000000000000000000000000000000000010000000000000000100000000000010000000000000000100L Using midpoint difference scheme to discretize multi-symplectic CNLKGZ system, we can get   qptccxaanlnlnlnl   12112121211    (5)   pqtddxbbnlnlnlnl   12112121211                                (6) 212121211   nlnlnl axpp                                                                                           (7) 212121211   nlnlnl bxqq                                                                                              (8) Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   56  212121121 nnnlnlctpp         (9) 212121121 nnnlnldtqq                                                                                                (10) 021211212112121121121 xvvxuuxeetgg nlnlnlnlnlnlnlnl    (11) 02121121121 xffthh nlnlnlnl                 (12) 212121211   nlnlnl ex                                                                                       (13) 212121211   nlnlnl fx                  (14) 212121121 nlnlnlgt                              (15) 212121121 nlnlnlht                                                                                           (16)         21212122112  nlnlxnlx uxpp                                                                 (17)       21212122112  nlnlxnlx vxqq                 (18)    2121212121212121ˆ,ˆ,ˆ,ˆ  nlnlnlnl ppqq   Eliminate a, and c from (5), using (7), (9) so we can get                  21 2121 2121 2121 2121 2321 2321 2321 232212121212321221211212112222nnnnnnnnnnnnnnnnnnnnnnnlnlnlqpqptpppxppp              (19)    Eliminate b, and d from (6), using (8), (10) so we can get                  21 2121 2121 2121 2121 2321 2321 2321 232212121212321221211212112222nnnnnnnnnnnnnnnnnnnnnnnlnlnlpqpqtqqqxqqq  (20)   Eliminate g, e, u, and v from (11), using (13), (15), (17), (18) so we can get Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   57                        022222212211221222212211221222212112122212121212321xqqqxpppxtnlxnlxnlxnlxnlxnlxnlnlnlnnnnnn               (21)      Eliminate h, and f from (12), using (14), (16) so we can get    3 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 2 12 22 20n n n n n nn n n l l lt x                                             (22) Multiply (20) with i. and adding Eq. (19) then we can get        3 2 1 2 1 21 2 1 2 1 21 2 1 2 1 22 12 21 2 1 2 1 2 1 23 2 3 2 1 2 1 22 22 21 1n n nn n nn n nl l ln n n nn n n nE E EE E Ex tE E                                   (23) Multiply (22) with i. And adding Eq. (21) then we can get          1 2 1 2 1 22 2 23 2 1 2 1 2 1 2 1 2 1 21 2 1 2 1 2 2 1 2 12 2 2(24)22 20n n nn n n n n nx x xn n n l l l l l lE E Et x x                                       3. Numerical simulation In this section, we present the numerical result of the Coupled Klein-Gordon-Zakharov system using the multi-symplectic integrator. Now we consider the CNLKGZ system 0,02xxxxttxxttEnnnEEEE with the initial value 00/41 1 0/41 2 01( , ) 2 sec ( )21( , ) 2 sec ( )2tV xtV xu x o r h r x D ev x o r h r x D e        (25) We take the time step ∆ t = 0.02 and a space step ∆x = 0.02, -0.3 ≤ x ≤ 0.3, D0 = 25, r1 = r2 = 1 and V0=1. In Fig. 1, the computation is done for 0 ≤ t ≤ 2.  Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   58  (a)        (b) Fig. 1. Simulation results of the interaction of the two waves 4. Conclusion: In this paper, the multi-symplectic formulation for the coupled Klein-Gordon-Zakharov system is presented. Numerical simulations are also reported. We observe that the multi-symplectic scheme well simulates the evolution of the solitons. It has advantage for the long time computing accuracy and preserving the energy conservation property. References [1] Ablowitz, M. J., & Segur, H. (1981). Solitons and inverse scattering transform. Philadelphia: SIAM. [2] Miura, R. M. (1996). Bäcklund transformation. New York: Springer-Verlag. [3] Liu, G. T., & Fan, T. Y. (2005). New applications of developed Jacobi elliptic function expansion methods. Phys. Lett. A, 345(1-3), 161-166. [4] Abdou, M. A. (2007). The extended F-expansion method and its application for a class of nonlinear evolution equations. Chaos Solitons and Fractals, 31(1), 95-104. 143 [5] Hirota, R. (2004). The direct method in soliton theory. Cambridge: Cambridge University Press. [6] Shang, Y. D. (2008). The extended hyperbolic functions method and exact solutions for long-short wave resonance equations, Chaos Solitons and Fractals, 36(3), 762-771. [7] Shang, Y. D., Huang, Y., & Yuan, W. J. (2008). New exact travelling wave solutions for the Klein-Gordon-Zakharov equations. Comp. Math. Appl., 56(5), 1441-1450. [8] He, J. H. (1998). An approximate solution technique depending on an artificial parameter: A special example. Commun. Nonlinear Sci. Numer. Simul., 3(2), 92-97. [9] Liu, Z. R., & Yang, C. X. (2002). The application of bifurcation method to a higher-order KDV equation. J. Math. Anal. Appl., 275(1), 1-12. Mathematical Theory and Modeling                                                             www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.3, No.3, 2013   59  [10] Liu, Z. R., Kayed, A. M., & Chen, C. (2006). Periodic waves and their limits for the Camassa-Holm equation. J. Bifurcation and Chaos, 16(8), 2261-2274. [11] Song, M., & Cai, J. H. (2009). Solitary wave solutions and kink wave solutions for a generalized Zakharov-Kuznetsov equation. Appl. Math. Comput., 217(4), 1455-1462.  [12] V. E. Zakharov, Collapse of Langmuir waves, Soviet Phys. JETP, 35 (1972), p. 908-914. [13] R. O. Dendy, Plasma Dynamics, Oxford University Press, Oxford, UK, 1990. [14] T.T. Liu, M.Z. Qin, Multisymplectic geometry and multi-symplectic Preissman scheme for the KP equation, JMP 43 (8) (2002) 4060–4077.  [15] T.J. Bridges, Multisymplectic structures and wave propagation, Math. Proc. Cambridge Philos. Soc. 121 (1997) 147–190. [16] Y.S. Wang, M.Z. 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A New Six Point Finite Difference Scheme for Nonlinear Waves Interaction Model

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A New Six Point Finite Difference Scheme for Nonlinear Waves Interaction Model

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