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.
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