In this paper, the improved Adomian decomposition method (ADM) is applied to the nonlinear Schrödinger’s equation (NLSE), one of the most important partial differential equations in quantum mechanics that governs the propagation of solitons through optical fibers. The performance and the accuracy of our improved method are supported by investigating several numerical examples that include initial conditions. The obtained results are compared with the exact solutions. It is shown that the method does not need linearization, weak or perturbation theory to obtain the solutions.
The study of optical solitons has been going on for the past few decades [
The dimensionless form of the generalized NLSE that is going to be studied in this paper is given by
Here, the dependent variable u is a complex valued function, while x and t are the two independent variables. The coefficients
The aim of this section is to obtain an exact bright, dark, and singular 1-soliton solution to this equation. The Ansatz method is used. In order to set up the starting point, the solitons are written in the phase-amplitude format as
and decomposing into real and imaginary parts lead to
speed v of the soliton as
For bright solitons, the starting hypothesis is
and
where, A represents the amplitude of the soliton and B is the inverse width of the soliton and v is the speed of the soliton.
Balancing principle yields
Substituting (2.5) into (2.4) we get
Setting the coefficients of the linearly independent functions
which will exist for the necessary constraints in place.
For dark solitons, the starting hypothesis is given by
For dark solitons the parameters A and B are free parameters. Substituting and applying Balancing principle yields
From coefficient of the
along with their respective constraints as indicated.
For singular solitons, the starting hypothesis is given by
From coefficient of
which will exist for the necessary constraints in place.
The NLS equation describes the spatio-temporal evolution of the complex field
The solution of a nonlinear Schrödinger equation will be reduced by using standard Adomian decomposition method [
where
mian decomposition method is assumed to be given by a series form
where the components
The
Operating on both sides of Equation (3.2) with the integral operator
Then following the Adomian decomposition method introduced by Wazwaz [
Setting
In this section, a reliable modification of the Adomian decomposition method decomposition method developed by Wazwaz [
Based on this, the modified recursive relation is formulated as follows
The choice of
In the new modification [
in this research, it is shown that if f consists of one term only, then scheme (4.1) reduces to relation (3.7). Moreover, if f consists of two terms, then relation (4.2) reduces to the modified relation (4.1).
Consider the nonlinear cubic Schrodinger equation (NLS) which has the general form
where
and boundary conditions
the initial boundary value problem (IBVP) (5.1)-(5.3) gives rise to soliton solutions in which the solution and its derivatives with respect to x vanish as
where
1) Standard Adomian Decomposition Method
Consider the initial condition
Using standard ADM the solution of the NLS equation is given by the following approximation;
The approximating Adomian decomposition method was tested to NLS equation for the single-soliton wave to the problems with boundary lines
2) Reliable Technique
Now we assume that the function f can be divided into two parts namely
In
Standard ADM | T |
---|---|
0.0000703381 | 0.1 |
0.000140626 | 0.2 |
0.000210864 | 0.3 |
0.000281052 | 0.4 |
0.00035119 | 0.5 |
0.000421277 | 0.6 |
0.000491314 | 0.7 |
0.0005613 | 0.8 |
0.000631235 | 0.9 |
0.00070112 | 1.0 |
Reliable ADM | T |
---|---|
0.0000704382 | 0.1 |
0.000140881 | 0.2 |
0.000211328 | 0.3 |
0.000281779 | 0.4 |
0.000352234 | 0.5 |
0.000422694 | 0.6 |
0.000491314 | 0.7 |
0.000563623 | 0.8 |
0.000634093 | 0.9 |
0.000704566 | 1.0 |
3) The New Modification
In the new modification, the process of dividing f into two components is replaced by a series of infinite components. The recursive relationship expressed in the form
Remark. When using the new modification for ADM the computations of the integrals will be simpler but we need a large number of components to get accurate results, which may lead to accumulation of round of error.
Consider the nonlinear Schrodinger equation (NLS)
Subject to the initial condition of the form,
1) The Standard ADM
So, we get the recurrent relation
We can calculate few terms as
The solution is
The behavior of the ADM solution obtained for different values of time is compared with the exact solution in
In
Remark: As it seen from
2) Reliable Technique
x | Absolute Error at t = 0.03 | Absolute Error at t = 0.1 | Absolute Error at t = 0.5 |
---|---|---|---|
0 | 3.370059347 × 10−8 | 0.0000041661332570 | 0.0025955039 |
0.1 | 3.370059347 × 10−8 | 0.0000041660562770 | 0.0025955040 |
0.2 | 3.381153649 × 10−8 | 0.0000041661206330 | 0.0025955039 |
0.3 | 3.374685170 × 10−8 | 0.0000041660544070 | 0.0025955038 |
0.4 | 3.377217790 × 10−8 | 0.0000041660937870 | 0.0025955040 |
0.5 | 3.379127106 × 10−8 | 0.0000041661050360 | 0.0025955038 |
0.6 | 3.370356064 × 10−8 | 0.0000041660713930 | 0.0025955040 |
0.7 | 3.375440712 × 10−8 | 0.0000041660511770 | 0.0025955038 |
0.8 | 3.373499667 × 10−8 | 0.0000041662206490 | 0.0025955036 |
0.9 | 3.376684765 × 10−8 | 0.0000041661306230 | 0.0025955037 |
1.0 | 3.370830758 × 10−8 | 0.0000041660782630 | 0.0025955036 |
We set
In
The behavior of the Reliable ADM solution is compared with the exact solution in
3) The New Modification
To apply the new modification we replace f by a series of infinite components. Then, we can calculate few terms as
where
The solution is
In
In
Reliable ADM | T |
---|---|
0.257913922 | 0.1 |
0.0448678059 | 0.2 |
0.0584708828 | 0.3 |
0.0687497541 | 0.4 |
0.079066386 | 0.5 |
X | Absolute Error at t = 0.03 | Absolute Error at t = 0.05 | Absolute Error at t = 0.1 |
---|---|---|---|
0 | +0.0000225 | +0.0001046 | +0.0008482 |
0.1 | +0.0000241 | +0.0001110 | +0.0009063 |
0.2 | +0.0000311 | +0.0001338 | +0.0010460 |
0.3 | +0.0000471 | +0.0001822 | +0.0013000 |
0.4 | +0.0000746 | +0.0002630 | +0.0016955 |
0.5 | +0.0001173 | +0.0003844 | +0.0022619 |
In our new calculation, the complex system given in Equation (1) in converted into a real system by writing
where
where
Applying the inverse operator
Assumes that, the nonlinear terms in (6.6) and (6.7) are represented by the following series
Following the decomposition analysis, we introduce the recursive relative
Adomian polynomials are calculated as follows
Similarly we can calculate
The method is applied to the two above examples.
1) Solution of Example 5.1 with IADM
We consider Schrodinger Equation (5.7) with its initial condition. In our calculation we will convert the complex equation given in Equation (5.7) into a real system by writing
As we explain above we get
The numerical results obtained with IADM are presented in
In
2) Solution of Example 5.2 with IADM
We consider following Schrodinger equation
With initial and boundary condition
As is explained above, the iterative relation is obtained as
Improve ADM | T |
---|---|
4.6052711 × 10−8 | 0.1 |
1.8421550 × 10−7 | 0.2 |
4.1449535 × 10−7 | 0.3 |
7.3689919 × 10−7 | 0.4 |
0.000001151433 | 0.5 |
Adomian polynomials are calculated as follows
Similarly we can calculate
The results are mentioned in
Improve ADM | T |
---|---|
1.6680003 × 10−7 | 0.1 |
0.000005331568 | 0.2 |
0.0000404691114 | 0.3 |
0.0001704381222 | 0.4 |
0.0005197377272 | 0.5 |
In this work, it is shown how the Adomian decomposition method and some of its modification can be adapted in order to be used to the nonlinear Schrodinger. The new method presented in this work has a powerful and easy use. The numerical technique is improved by decomposition of the nonlinear operator. In applying the improved Adomian decomposition Method (IADM) to the nonlinear Schrodinger equation, it is found that the method gives accurate results with lesser computational effort as compared with other modification.
The authors declare that there is no conflict of interest regarding the publication of this paper.
Al-Shareef, A., Al Qarni, A.A., Al-Mohalbadi, S. and Bakodah, H.O. (2016) Soliton Solutions and Numerical Treatment of the Nonlinear Schrodinger’s Equation Using Modified Adomian Decomposition Method. Journal of Applied Mathematics and Physics, 4, 2215-2232. http://dx.doi.org/10.4236/jamp.2016.412215