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Research Journal of Applied Sciences, Engineering and Technology 4(19): 3834-3837, 2012 ISSN: 2040-7467 © Maxwell Scientific Organization, 2012 Submitted: April 27, 2012 Accepted: May 13, 2012 Published: October 01, 2012 Split-Step Multi-Symplectic Method for Nonlinear Schrödinger Equation 1 Zainal Abdul Aziz, 2Nazeeruddin Yaacob, 3Mohammadreza Askaripour Lahiji and 4 Mahdi Ghanbari 1, 2, 3 Department of Mathematics, Faculty of science, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia 4 Department of Mathematics, Islamic Azad University, Khorramabad Branch Abstract: Multi-symplectic methods have recently been considered as a generalization of symplectic ODE methods to the case of Hamiltonian PDEs. The symplectic of Hamiltonian systems is well known, but for Partial Differential Equation (PDEs) this is a global property. In addition, many PDEs can be written as Multisymplectic systems, in which each independent variable has a distinct symplectic structure. Also, Their excellent long time behavior for a variety of Hamiltonian wave equations has been proposed in a number of numerical studies. In the study, a new type of multi-symlectic integrators, which is used for solving Nonlinear Schrödinger Equation (NLS) has been demonstrated. Keywords: Conservation law, multi-symplectic scheme, schrödinger equation, split-step method INTRODUCTION First of all, it is a good idea to recall symplectic structure and Hamiltonian systems. Moore and Reich (2003) studied a useful method for understanding discretization error in the numerical solution of ODEs which served to compare the system of ODEs with the modified equation obtained through backward error analysis. In another study, (Hong and Li, 2003; 2006) discussed the multi-symplecticity and energy and momentum conservation laws of the nonlinear equation and also presented a definition for multi-symplecticity of Runge-Kutta discretization for HPEDs. They applied the Multi-Symplectic Runge-Kutta (MSRK) methods for nonlinear Dirace equation in relativistic quantum physics, based on a discovery of multi-symplecticity of the equation. In a recent study, Ascher and Mclachlan (2004) stated that the discretization of multi-Hamiltonian in space and time with partitioned Runge-Kutta methods gives rise to a system of equations that formally satisfy a discrete multi-symplectic conservation law. According to Bridges and Reich (2001), one of the great challenges in the numerical analysis of Partial Differential Equation (PDEs) is the development of robust stable numerical of Hamiltonian PDEs. Mclachlan and Quispel (2002) claimed that although Runge-Kutta and partitioned Runge-kutta methods are known to formally satisfy multisymplectic conservation laws when applied to the multiHamiltonian PDEs, they do not always lead to welldefined numerical methods. They considered the case study of the nonlinear Schrödinger equation. According to Bridges and Hydon (2005), many well-known partial differential equations can be written as multi-symplectic Systems such systems have a structural conservation law form which scalar conservation laws can be derived. They discussed the relationship between Potential Vorticity (PV) and the symplectic from is explored, for the shallowwater equations governing Lagrangion particle path. Starting with the symplectic form, the PV is found by the pull-back operation to reference space. In this study, we use some definitions that were proposed. A new kind of multi-symplectic integrators is presented, which is called split-step multi-symplectic methods. Moreover, in this study, the Perissman scheme and the Euler midpoint scheme are applied. SYMPLECTIC MAP A linear map in R2 is symplectic if: ATJA = J where, ⎡ 0 1⎤ J=⎢ ⎥ ⎣ − 1 0⎦ Nonlinear maps are symplectic if their linearization is symplectic (Berland, 2005; Wang and Qin, 2002). Hamiltonian systems: JYt = LH (y) Corresponding Author: Mohammadreza Askaripour Lahiji, Department of Mathematics, Faculty of science, Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia 3834 Res. J. Appl. Sci. Eng. Technol., 4(19): 3834-3837, 2012 where, y plays the role of z, H (y) is the Hamiltonian (Bridges and Reich, 2001; Bridges et al., 2006). The description of symplectic and multi-symplectic structure is written in terms of differential forms. Definition (1-form):A 1-form is map T1 : Rn ÷ R For example, if g = (x, y, z) in R3, then T1 (g) = c, c constant The Eq. (1) can be written as a multi-symplectic, where Q is complex. Taking Q = p +iq and separating the real and imaginary components of NLS allows the PDE to be written in the form multi-symplectic Hamiltonian system: Mzt+Kzx = L zS(z) (2) where, Definition (2-form): An exterior form of degree 2 is a map on pairs of vectors: ⎡0 ⎡ p⎤ ⎢1 ⎢q⎥ ⎥ ⎢ z= ,M = ⎢ ⎢0 ⎢ v⎥ ⎢ ⎢ ⎥ ⎣0 ⎣ w⎦ 0 1 ⎡0 ⎢0 0 0 K= ⎢ ⎢− 1 0 0 ⎢ ⎣ 0 −1 0 T2 : Rn × R ÷ R which is bilinear and skew-symmetric are shown below respectively: T2(81g1 + 82g2, g3) = 81T2 (g1, g3) + 82T2(g2, g3) T2(g1, g3) = ! T2(g2, g1) and S=− MULTI-SYMPLECTIC STRUCTURE Given the multi-symplectic of the Partial Differential Equation (PDE) Mzt +Kzx = LzS(z) where, z (x, t) , Rd and M, K , Rd×d are skew-symmetric matrices and S : Rd ÷ R is a smooth function of the phase variable (Ryland, 2007; Chen and Qin, 2001). The multisymplectic structure is given by two 2-forms: 1 2 ( p + q 2 ) − 21 (v 2 + w2 ) 2 S(z) = S1(z) + S2(z) (4) The multi-symplectic (1) splits into sub-multisymplectic systems: Mzt + K j zx = ∇ x S j ( z ), j = 1, 2 Conservation of multi-symplecticity: Lemma: (3) Now, we want to do the splitting multi-symplectic formulation (2) through the nonlinear Schrödinger. Decompose the spatial symplectic structure matrix into K = K1 +K2 and Hamiltonian function into: ω = dz ∧ Mdz and κ = dz ∧ Kdz (5) It is easy to show that the subsystem (5) satisfy Multi-Symplectic Conservation Law (MSCL), namely: Tt + 6x = 0 Tt+kjx = 0, Proof: j = 1, 2 (6) where, the new 2-forms are: ωt + κ x = dzt ∧ Mdz + dz ∧ Mdzt + dzz ∧ Kdz + dz ∧ Kdzx kt = dz v Kjdz, = − ( Mdzt + Kdzx ) ∧ dz + dz ∧ ( Mdzt + Kdzx ) j = 1, 2 (7) It is clear that 6 = 61 + 62 , in fact: = − Szz dz ∧ dz + ∧ Szz dz =0 κ = dz ∧ Kdz = dz ∧ ( K 1 + K 2 )dz (Using Leibniz rule, skew-symmetry, the differential equation and the fact that Szz is symmetric) Split-step multi-symplectic method: Let us consider the Nonlinear Schrödinger equation (NLS): iQt + Qxx + 2|Q|2 = 0 − 1 0 0⎤ 0 0 0⎥ ⎥, 0 0 0⎥ ⎥ 0 0 0⎦ 0⎤ 1⎥ ⎥ 0⎥ ⎥ 0⎦ = dz ∧ K 1dz + dz ∧ K 2 dz = κ 1 + κ 2 (8) In addition, both of the subsystems (5) conserve the total symplecticity: l ϖ ( t ) = ∫ ω ( x , t )dx (1) o 3835 (9) Res. J. Appl. Sci. Eng. Technol., 4(19): 3834-3837, 2012 Fig. 1: The central box of the perissman scheme In this study, we introduce linear-nonlinear splitting for nonlinear Schrödinger equation. Let us first rewrite Eq. (1) in the form: iψ t = ( L + N )ψ In fact, the nonlinear multi-symplectic system (14) reduces to an infinite dimensional Hamiltonian: (10) ∂ ∂x 2 pt = !2(p2 + q2)q (17) qt = !2(p2 + q2)p (18) 2 L= (11) N = − 2ψ 2 (12) iψ t = Lψ = ψ xx (13) 2 iψ t = Nψ = − 2 ψ ψ ⎡0 ⎢0 K1 = ⎢ ⎢− 1 ⎢ ⎣0 0 0 0 0 S 1( z) = − 1 0 0 0 0⎤ ⎡0 0 0⎥ 2 ⎢ 0 0 ⎥, K = ⎢ ⎢0 0 0⎥ ⎥ ⎢ 0⎦ ⎣0 − 1 1 2 (v + w 2 ) 2 (14) 0 0 0 0 The linear sub problem (13) and nonlinear sub problem (14) can be approximated by multi-symplectic integrators. We consider the central box scheme to a discrete linear sub problem (13). One of the most popular multisymplectic is the Perissman scheme, which corresponds to mid-point discretization in space and time variables (Fig. 1). The Preissman scheme for (2) is given by: 0⎤ 1⎥ ⎥ 0⎥ ⎥ 0⎦ n+ 1 2 ( p + q2 ) 2 n+ 1 2 ⎛ n+ 1 ⎞ = ∇ z S ⎜⎜ z 21 ⎟⎟ (19) ⎝ m+ 2 ⎠ with (15) z n m+ and S 2 ( z) = − 1 z n +1 − z n z 2 − zm M m+ 1 m+ 1 + K m+ 1 ∆t ∆x 1 2 n + 21 (16) zm 3836 zmn + zmn+1 = 2 (20) n n +1 zm + zm 2 (21) = Res. J. Appl. Sci. Eng. Technol., 4(19): 3834-3837, 2012 1 2 1 m+ 2 z n+ = zmn + zmn+1 + zmn+1 + zmn++11 4 (22) where, )t, )x are time and space respectively and znm is an approximation to z (n)t, m)x) For the nonlinear sub problem (14), we first discretized it in space and take a finite dimensional Hamiltonian system: ∂p j = − 2 p2j + q 2j q j ∂t ( ) j = 1, 2, 3, ..., N ∂q j = − 2 p2j + q 2j p j ∂t ( ) j = 1, 2, 3, ..., N (23) (24) And the finite-dimensional Hamiltonian system (23) and (24), we use the Euler midpoint scheme that is symplectic: z n +1 − z n ⎛ n+ 1 ⎞ = j∇ z H ⎜ z 2 ⎟ ⎝ ⎠ ∆t (25) CONCLUSION We have shown the symplectic of Hamiltonian system and proposed it for multi-symplectic integrators. Since we believe that Multi-symplectic integrators for PDEs are able to catch more physical features of the system than symplectic integrators. The Preissman box scheme and Euler midpoint scheme have been applied to the multi-symplectic Hamiltonian PDEs. Furthermore, we have presented a new kind of Multi-symplectic integrators, which is called split-step Multi-symplectic method. REFERENCES Ascher, U.M. and R.I. Mclachlan, 2004. Multi symplectic box scheme and Korteweq-de Vries equation. Appl. Numer. Math., 48: 255-269. Berland, H., 2005. Introduction to Multi-Symplectic Integrators. Department of Mathematical sciences, NTNU, Norway. Bridges, T.J. and S. Reich, 2006. Numerical methods for hamiltonian PDEs. J. Phys. A: Math. Gen., 39(Article Number 5287). Bridges, T.J. and S. Reich, 2001. Multi-symplectic integration: Numerical schemes for Hamiltonian PDEs that conserve symplecticity. Phys. Let. A., 284: 184-193. Bridges, T.J. and P.E. Hydon, 2005. 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