numerical simulation of nonlinear schrödinger systems: a new conservative scheme

NORTH- HOLLAND

Numerical Simulation of Nonlinear SchrSdinger Systems: A New Conservative Scheme

Zhang Fei* Victor M. Pdrez-Garcia, and Luis Vg~zquez

Departamento de Fisica Tedrica I Facultad de Ciencias Fisieas Universidad Complutense E-280~0 Madrid, Spain

Transmitted by John Casti

ABSTRACT

We present a new numerical scheme for nonlinear SchrSdinger type equations. The scheme conserves the energy and charge of the systems and it is linearly implicit. Numerical experiments for several physical problems show that our scheme is stable, accurate, and efficient.

1. INTRODUCTION

As it is well known, the cubic nonlinear SchrSdinger equation (NLS)

iw t + w~x + alWl2W = o,

w(x , o) = Wo(x), (i)

where a = const, and W(x, t) being a complex function, has a wide applica- tion in physics. It arises as an asymptotic limit of slowly varying dispersive wave envelope in nonlinear medium such as nonlinear optics, water waves, plasma physics, biomolecular dynamics, etc. (see, e.g., [1-4 D. In particular recently, increasing attention has been paid to the analysis of nonlinear pulse propagation in optical fibers that can be described in the framework of the NLS (1).

In principle, (1) can be solved exactly by using the inverse scattering transform [5]; however, the corresponding solutions are not very explicit except for the special cases of soliton solutions. Moreover, in real physical

* Research School of Chemistry, The Australian National University, Canberra ACT 2601, Australia.

APPLIED MATHEMATICS AND COMPUTATION 71:165-177 (1995) (~) Elsevier Science Inc., 1995 0096-3003/95/$9.50 655 Avenue of the Americas, New York, NY 10010 SSDI 0096-3003(94)00152-T

166 Z. FEI ET AL.

systems, existence of driving forces or dissipations may give rise to additional perturbations to the ideal model (1), and thus make it intractable analytically. In order to understand the dynamics of the complicated per- turbed soliton models one has to use numerical simulations. It must be noted that not all schemes can give reliable numerical results, and inap- propriate discretization may induce unphysical "blow-up" and "numerical chaos" (see, e.g., [6-9]).

On the other hand, the NLS (1) has an infinite number of conserved quantities, among which are the energy and the charge:

E = IW~]2dx- -~ IW:dx, (2) OD O0

/5 Q -- Iwl 2 dx. (3) o ~

These two conservation laws can guarantee the boundness of the solution of the NLS (1) [10]. Therefore, it is natural to construct some numerical schemes that have discrete analogues of the conservation laws.

So far, several numerical schemes have been proposed for the NLS (1), based on either finite differences, finite elements, or spectral methods (see, e.g., [11-17]). In particular, considerable effort has been devoted to the con- struction of the conservative schemes which have discrete analogues of the continuous energy and charge defined by (2) and (3). Based on the method of Strauss and V~zquez [18], Delfour et al. [11] proposed the first conservative scheme for the NLS. Other conservative schemes were presented in [14-16]. The stability and convergence property of some schemes were ana- lyzed in [13, 15, 16]. By extensive numerical experiments it has been found that the conservative schemes can perform better than the nonconservative ones, and the latter schemes may easily show nonlinear blow-up [17].

However, the conservative schemes proposed in the previous papers [11, 14-16] for (1) are globally nonlinear implicit, i.e., at each discrete time step a set of nonlinear algebraic equations has to be solved, requiring much computer time. Moreover, there is the additional problem of selecting the appropriate iteration procedure to solve the nonlinear algebraic equations in order to guarantee the convergence.

In the present paper, we propose a conservative scheme that is globally linearly implicit, which means that at each discrete time level we only need to solve a set of linear algebraic equations. As a consequence our scheme is faster and simpler than the nonlinearly implicit ones previously proposed in [11, 14-16]. In Section 2 we present the scheme and discuss its properties. We show rigorously that our scheme is stable and convergent and that it will not yield "blow-up." Section 3 is devoted to the numerical tests of our

Conservative Scheme for NLS 167

scheme for one-, two-, and three-soliton problems. It is found that the new scheme is faster and more accurate than the other conservative schemes. In Section 4 we discuss a generalization of our scheme and we apply it to a model of laser optics including dissipative effects. Satisfactory numerical simulation results are obtained. Finally, in Section 5 we conclude our paper with some remarks.

2. THE NUMERICAL SCHEME

We propose the following conservative numerical scheme for (1),

W n+l _ W~ -1 i I

2T 1 (. n-I-1 __ 2w~+l w~+l - -t- w n-l~

2 h 2 i,tul_t_ 1 -t- -t- WT.~l 1 - - 2W~ -1 l--1 ]

a (wt+x 2 lwFl~ + w~-i), (4)

~o = Wo(Zh),

where T and h are the spatial and temporal stepsizes, respectively, and w~ =- w(lh, nT).

The scheme (4) is consistent with the continuum NLS equation and the local truncation error is O(~ 2 + h2). The advantage of this scheme with respect to those of [11, 14-16] is that it is globally linear implicit, which means that at each discrete time level we only need to solve a set of linear algebraic equations to get w~ +1. However, the scheme is not selfstarting, in the sense that the function values w~ have to be provided by other scheme such as Cauchy iteration or the Crank-Nicholson linear implicit scheme with a smaller time step (for instance 7-/10) . In our numerical simulation we use the Crank-Nicholson linear implicit scheme [12] with a time step 7-/10 t o calculate the value w(lh, T / 1 0 ) , and then we use repeatedly the scheme (4) with timestep 7-/10 until we get the values w~. Then we go ahead to simulate the problem using the scheme (4) with timestep 7-.

Most importantly, the scheme (4) has constant energy and charge, which are the discrete analogous for (2) and (3):

z n = Z Wl+l -h +

a Z hlw~+1121W~12' (5) 2

l

Q ° : E + • l

168 Z. FEI ET AL.

The existence of the above discrete conservation laws is essential to guarantee the stability and convergence of the numerical scheme. First let us briefly prove the convergence of the scheme. Assume that w'~(x) = W'~(x) + ~-~(x) (here x = lh), where Wn(x) is the exact solution, wn(x) is the numerical solution of (4), and ~-~ (x) is the error. From the conservation laws (6) it is obtained

Ilwnll = ~ IIW°ll 2 + IlWlll 2,

where II " II denotes the L2-norm of the function defined as follows:

(7)

IIw~ll ~ ~ (wn,w n) = h ~_w?'~w?. (8) l

Therefore, the error can be estimated as follows:

II~nll ~ ~ IIw n - Wnll z ~ 2(llW°ll 2 + IlWXll ~) + 211W°ll 2. (9)

Moreover, the error functions satisfy the following finite difference equation:

2T 1 [=.=.n+l _ 2~?-I-1 - - -

__ 2 h 2 k W l + l ..~ ~?--F? _~_ W?~ll 2 ~ ? - 1 ..~_ ~?...-?)

a ~-~n 2[.zT~n+l - i ~ ~ + ~?-~) + G? + F?,

where F~ is the truncation error which is of order O(T 2 + h2), and

a / u l ~ + l a ? = 2 ~.. , + w ? - l ) (IW?l~ - l~'tl ~)

+ ~ + ~ ? - 1 ) ( l~rl ~ - Lwrl2).

Then it is obtained

(10)

(11)

IIG"II ~ MIlI~"II + M2(llwn+lll-[-I1~"-111), (12)

where M1 and M2 are two constants only depends on the initial data, w ° and w~.

Multiplying (10) by ( ~ + 1 + ~-~-1),, summing over j , and taking the imaginary part, we obtain

2T = im((Gn,~*n+x + ~ , n - 1 ) + (Fn,~*n+l + ~*n-1)). (13)


Combining (9), (12), and (13) we get

n + l

ll~"+lll 2 < II~ll 2 + II~lll 2 + r c ~ I I~ l l 2 + ~ IIFmll 2, (14)

where C is a constant depending on the initial data. Let us assume that T is small enough such that 1 - CT --= A > 0; then we may use the theory of error estimate developed in [15, 19, 20] to obtain the result

1 ( ~ ) ( C ( n + l ) v ) (15) jj~n+xjj2 _< A jjw--Ojj2%jj~xjj2+r [Jf'~JJ 2 exp A " r n = l

This means that the numerical error is bounded by the initial and truncation errors, so the scheme is convergent.

Finally, we would like to point out that the scheme (4) will never show numerical blow-up given the conservations of the energy and charge. This can be proven by estimating the C ~ bound of w~ with the method found in [21]. We find that the numerical solution is bounded by:

Jw'~J 2 <_ 2Q°/L + 2a(Q°) 2 + x/4a2(Q°) 4 + 2a(Q°)3/L + Q°E°, Vl, n,

(16)

where E ° and Q0 are defined by (5) and (6) and L is the length of the spatial interval.

3. NUMERICAL TESTS

In this section we will present the numerical simulation results with the scheme (4) discussed in Section 2.

The following initial conditions,

W (x, O) = 2TlV~e2iXz sech{2r/(x - x0)}, (17)

/ r ~ 2i z W(x,O) --- 2 r / i r a e x, sech{2rll(X - xl)}

] ' 2 2i x +2~72Va e ×2 sech{(2772(x- x2)} (18)

W(x, O) = sech(x - xl) (19)

are used to check the accuracy of our method. Unless the contrary is stated the standard value for the nonlinear constant is a = 2.0.

170 Z. FEI ET AL.

~5 FIG. 1. Propagation of a single soliton.

Initial da ta (17) is the usual one-soliton solution which is integrated without problems by most methods. A result of an integration with ~] = 0.5, X = 0.5, x0 = 20.0 over the spatial interval x E [0, 60] and t ime interval 0 < t < 12 with integration parameters T = 0.02, h = 0.1 is presented in Figure 1.

The expression in (18) is an initial data for a pair of solitons with different amplitudes and velocities and it is appropriate for the simulation of soliton collision (assuming the soliton centers are initially set far away from each other). This is another s tandard test of the accuracy of an integration method. For example in [17] a collision of two solitons with parameters 711 -- 712 = 0.5, X1 = 0.25, X2 : 0.025, xl = 20.0, x2 = 45.0 in the region x E [0.0, 100.0] was simulated using ~- = 0.125, h -- 0.25. Some of the methods in [17] failed to give an accurate solution, but the bet ter one, which used a finite difference in space and an implicit midpoint rule to solve the resulting set of ODEs did converge for these stepsizes. Due to the nonlinear implict character of those methods the conservation of the charge was assured up to 10 -4 . At each t ime step, an average of three linear systems had to be solved because of the nonlinear nature of the system of equations.

With T ---- 0.125, h ---- 0.25, our method also converges for the case of two-soliton collision as in Figure 2, and the numerical results are accurate enough. The charge and the energy are exactly conserved in the simulation. Also the integration t ime is shorter by a factor about four because only a simple linear system (a tridiagonal matr ix is involved) per step needs to be solved.


%

c ~

FIG. 2. Interaction of two solitons.

The simulation with the initial data (19) is usually considered to be a more difficult "quality" test for numerical schemes because of the ap- pearance of large spatial and tempora l gradients in the solution. For a = 2N 2 (N = 2, 3 , . . . ) Miles [22] showed that (19) corresponds to a bound state of N solitons. For the case a -= 18 it has been found in [17] that the spatial stepsize must be smaller than h < 0.03125 in order to get relevant numerical results. The best method in [17] with T = 0.00625 requires five iterations in average. With the same steps our method converges without problems while enhancing the speed by a factor five.

Two finite element method used by Herbst et al. [14] (called method I and III in their paper) resembled the exact solution for bigger spatial steps h = 0.1 with ~- = 0.005 (an increase in the value of 7- led to quantitatively important differences with the exact solution). For these stepsizes our method failed to represent the solution accurately; however, halving the spatial step was enough to achieve good convergence as in Figure 3 with significantly less computational effort than the method by [14] despite their use of a bigger spatial step (because they needed four iterations per step to solve the nonlinear system of equations). So an enhancement of the speed by a factor two can be achieved with our method, with the additional advantage of the conservation of energy and charge.

No "blow up" has been observed in our numerical simulations even in the cases of very rough discretizations. This agrees totally with our analytical results established in the previous section (cf., (16)).

172 Z. FEI ET AL.

1

~,(Z)

~b

FIG. 3. Simulation of the bounded state of three solitons.

4. EXTENSION TO A NONCONSERVATIVE CASE

Let us consider now the following family of nonconservative nonlinear SchrSdinger equations:

iWt + Wx~ + alWI2W + iF(IWI)W = O. (20)

Now the charge is not conserved but satisfies the equation

d-td / I W I 2 dx = -2 / F(IWI)IWI 2 dx. (21)

The family (20) is of considerable interest in nonlinear optics, where it appears in many cases with different forms (see, e.g., [23] for applications


in other areas). We will concentrate on two simple examples to illustrate how our scheme works:

• F ( I W I ) ---- v. This case simply corresponds to the dissipative nonlinear Schrbdinger equation. It appears in many physical systems, for example, in the analysis of propagation of light pulses in absorptive nonlinear optical fibers [24] or in the propagation of beams in Kerr media [25]. In this case, the equation for the charge variation (21) can be exactly integrated, the solution being an exponential decay:

Q(t) = Qoe -2yr. (22)

• F ( I W I ) = v - glFI 2. This term is obtained (for example) in the adiabatic reduction of the Maxwell-Bloch equation in one transverse spatial dimension in the near threshold approximation [26]. In this case the evolution equation for the charge (21) cannot be exactly integrated, but in the case in which Q(0) < 1 and g <_ v it is easy to find that the charge must decrease monotonically and tend to an exponential decay in the asymptotic regime.

We will discretize the nonconservative term in the following way:

+ 2 . (23)

By inserting this discretization into the evolution equation, the following discrete analog of the continuous equation is found:

E = - 2 F(Iw~l ) (24) r 2 k k

which is a consistent discretization of the underlying continuous equation (21).

The purely dissipative case was discussed in [11, 27]. We applied our method to simulate the evolution of the soliton (17) with parameters 77 = 0.5, X = 0.5 in the region x E [0.0, 100.0] with spatial step h = 0.1 and time step T ---- 0.05. Figure 4 shows results for v -- 0.1 and some different dissipations. The charge had an exponential decrease (Figure 4b) as predicted by (22).

For the pumped-dumped case we run the case with initial soliton parameters ~ - 0.5, X -- 1.0, x0 -- 15.0, steps ~- -- 0.05 and h = 0.1, and some different pumping and damping constants. The results are perfectly consistent with the qualitative predictions about the nonconservative equations as in Figure 5.

174 Z. FEI ET AL.

e(t)

1.0

0.8

0.6

0.4

0.2

0.0

(b)

o 2b #o t

60

FIG. 4. (a) Propagation of a single soliton with dissipation (v = 0.1) and (b) t ime variation of the charge. An exponential decrease is clearly observed.


1.0

0.8 ~',,,

\ \ ',, 0.6 \\,,,

o(t) ,\',, \\',, 0.4 \\,,,

\ \ ' , \ ~ ' , ,

0.2 \~',...

I I t I t

0.0 0 20 40 6O Time

FIG. 5. Time evolution of the charge in the pumped case. The dissipation is set to v = 0.1. - - - - corresponds to g -= 0.0 (no pumping), - - to g ---- 0.05, and - - - to g ---- 0.1. The curves are monotonically decreasing, tend to the same exponential decay in the long time regime, and are ordered by the value of the pumping (the greater the pumping the higher the curve over the time axis) as predicted in the text.

5. C O N C L U D I N G R E M A R K S

In conclusion, we have p resen ted a s imple, conservat ive finite difference scheme for nonl inear Schr6dinger sys tems. T h e new scheme shows some clear advan tages over the prev ious ly p roposed in tegra t ion me thods . In par - t icular , the new scheme is easier to be encoded and requires less c o m p u t e r t ime. I t has been proven to be convergent and s table . No numer ica l blow- up can a p p e a r due to the conserva t ion of b o t h the energy and charge of the sys tem.

Our scheme can be genera l ized in m a n y phys ica l ly in te res t ing cases. In the p resen t p a p e r we have discussed the d a m p e d - p u m p e d nonl inear SchrSdinger equa t ion , and we have ob t a ined sa t i s fac to ry numer ica l resul ts . In fact , t he scheme can be app l i ed to s t u d y the d y n a m i c s of the NLS wi th o the r t ypes of p e r t u r b a t i o n s ar is ing in real phys ica l sys tems.

176 Z. FEI ET AL.

Finally, we would like to point out that our scheme can also be straight- forwardly generalized to the case of two-componentNLS, which appear in the description of the double-mode optical fiber, or the propagation of pulses in a nonlinear dualco~e directional coupler (see, e.g., [28, 29]). In this case the main features of our numerical scheme can be preserved. Thus we believe that the new scheme is also recommended to study the coupled nonlinear SchrSdinger systems.

Zhang Fei acknowledges the Ministerio de Education y Ciencia of Spain for a research fellowship (F.P.I.PG89). V. M. Pdrez-Garcla thanks the Univereidad Complutense for the support under a predoctoral fellowship. This work has been also partially supported by the Comision Interministerial de Ciencia y Tecnologia of Spain under Grant Nos. MATgO/05~ and PB92-OP26.

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numerical simulation of nonlinear schrödinger systems: a new conservative scheme

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