- Research
- Open Access
Conservative finite difference schemes for the chiral nonlinear Schrödinger equation
- Mohammad S Ismail^{1}Email author,
- Khalil S Al-Basyouni^{1} and
- Ayhan Aydin^{2}
- Received: 29 January 2015
- Accepted: 17 May 2015
- Published: 3 June 2015
Abstract
In this paper, we derive three finite difference schemes for the chiral nonlinear Schrödinger equation (CNLS). The CNLS equation has two kinds of progressive wave solutions: bright and dark soliton. The proposed methods are implicit, unconditionally stable and of second order in space and time directions. The exact solutions and the conserved quantities are used to assess the efficiency of these methods. Numerical simulations of single bright and dark solitons are given. The interactions of two bright solitons are also displayed.
Keywords
- Soliton Solution
- Finite Difference Scheme
- Discrete Analog
- Dark Soliton
- Bright Soliton
1 Introduction
The conserved quantities (6)-(9) using dark soliton solution (4) are not well defined [5] due the nonzero boundary condition as \(|x|\rightarrow \infty\). To overcome this difficulty, we present a Robin-type boundary condition [5], which will lead us to a modified form of the conserved \(I_{1}(t)\) which is conserved exactly as we will see in the Numerical results section.
There are many theoretical and numerical studies in the literature about the nonlinear Schrödinger equations (NLS). Most of these works are motivated to study single NLS and coupled NLS (see [6–9] and references therein). However, to the authors’ knowledge, there are few numerical studies for the CNLS equation. In this paper we have derived three conservative finite difference schemes for the CNLS equation. The paper is organized as follows. In Section 2, three conservative schemes are proposed for the numerical solution of the chiral NLS. In Section 3, theoretical and numerical conservation properties are proved. Accuracy of the proposed schemes is studied in Section 4. Stability analysis is given in Section 5. Numerical results are presented in Section 6. Finally, some conclusions are drawn in Section 7.
2 Numerical methods
We will consider the numerical solution of the nonlinear system (10)-(11) in a finite interval \([x_{L},x_{R}]\). We assume \(x_{m}=x_{L}+mh\), where \(m=1,2,\ldots,M-1\), and h is called the space grid size, also we assume \(t_{n}=nk\), k is the time step size. We denote the exact and numerical solutions at the grid point \((x_{m},t_{n})\) by \(\mathbf{w}_{m}^{n}\) and \(\mathbf{W}_{m}^{n}\), respectively. In this work we will present the following numerical schemes for solving (12).
2.1 Scheme 1 (nonlinear implicit scheme)
2.2 Scheme 2 (linearly implicit scheme)
2.3 Scheme 3 (linearly implicit scheme 3)
The resulting system in (20) and (21) is a linear block tridiagonal system for the unknown numerical solution \(\mathbf{W}^{n+1}\), which can be easily solved by Crout’s method. The scheme conserves the discrete analog of the conserved quantity (6). The accuracy of Scheme 3 is of second order in time and space, it is unconditionally stable according to the von Neumann stability analysis. The scheme is a three-level scheme and the solutions at \(t=0\) and \(t=k\) are required in order to get the solution at \(t=nk\), \(n=2,3,\ldots\) , the same procedure in Scheme 2 can be adopted.
3 Conserved quantity
To prove that the proposed schemes preserve the discrete analog of invariant (6), we need the following lemma [9, 10].
Lemma 1
4 Accuracy of Scheme 2
The first quantity in the right-hand side of Eq. (38) is zero by the differential system under consideration, which means that Scheme 2 is of second order in space and time. Similar analysis can be done for the other schemes.
5 Stability of the proposed schemes
6 Numerical results
In this section, we will test the efficiency of the numerical schemes presented in this work, by considering different numerical tests. Trapezoidal rule is used to calculate the conserved quantities.
6.1 Bright soliton solution
Conserved quantities (Scheme 1: bright soliton, \(\pmb{A=0.5}\) , \(\pmb{\lambda=0.5}\) , \(\pmb{v=0.5}\) )
t | \(\boldsymbol{I_{1}}\) | \(\boldsymbol{I_{2}}\) | \(\boldsymbol{I_{3}}\) | \(\boldsymbol{I_{4}}\) | \(\boldsymbol{L_{\infty}}\) | \(\boldsymbol{L_{2}}\) |
---|---|---|---|---|---|---|
0 | 1.414214 | 1.413993 | 2.945836 | 3.045932 | - | - |
5 | 1.414214 | 1.413988 | 2.945829 | 3.045923 | 0.00034 | 0.00202 |
10 | 1.414214 | 1.413975 | 2.945808 | 3.045898 | 0.00082 | 0.00458 |
15 | 1.414214 | 1.413948 | 2.945761 | 3.045846 | 0.00198 | 0.01076 |
20 | 1.414214 | 1.413842 | 2.945560 | 3.045636 | 0.00480 | 0.02581 |
Conserved quantities (Scheme 2: bright soliton, \(\pmb{A=0.5}\) , \(\pmb{\lambda=0.5}\) , \(\pmb{v=0.5}\) )
t | \(\boldsymbol{I_{1}}\) | \(\boldsymbol{I_{2}}\) | \(\boldsymbol{I_{3}}\) | \(\boldsymbol{I_{4}}\) | \(\boldsymbol{L_{\infty}}\) | \(\boldsymbol{L_{2}}\) |
---|---|---|---|---|---|---|
0 | 1.414214 | 1.413993 | 2.945836 | 3.045932 | - | - |
5 | 1.414214 | 1.413987 | 2.945828 | 3.045922 | 0.00034 | 0.00202 |
10 | 1.414214 | 1.413974 | 2.945806 | 3.045896 | 0.00082 | 0.00458 |
15 | 1.414214 | 1.413946 | 2.945758 | 3.045843 | 0.00198 | 0.01076 |
20 | 1.414214 | 1.413840 | 2.945557 | 3.045633 | 0.00480 | 0.02582 |
Conserved quantities (Scheme 3: bright soliton, \(\pmb{A=0.5}\) , \(\pmb{\lambda=0.5}\) , \(\pmb{v=0.5}\) )
t | \(\boldsymbol{I_{1}}\) | \(\boldsymbol{I_{2}}\) | \(\boldsymbol{I_{3}}\) | \(\boldsymbol{I_{4}}\) | \(\boldsymbol{L_{\infty}}\) | \(\boldsymbol{L_{2}}\) |
---|---|---|---|---|---|---|
0 | 1.414214 | 1.413993 | 2.945836 | 3.045932 | - | - |
5 | 1.414214 | 1.413987 | 2.945828 | 3.045923 | 0.00034 | 0.00203 |
10 | 1.414214 | 1.413974 | 2.945807 | 3.045898 | 0.00082 | 0.00460 |
15 | 1.414214 | 1.413947 | 2.945759 | 3.045846 | 0.00198 | 0.01077 |
20 | 1.414214 | 1.413841 | 2.945558 | 3.045636 | 0.00480 | 0.02582 |
6.2 Dark soliton solution
Dark soliton (Scheme 1)
t | \(\boldsymbol{L_{\infty}}\) | \(\boldsymbol{L_{2}}\) | \(\boldsymbol{I_{11}}\) ( 47 ) |
---|---|---|---|
0 | 0.000000 | 0.000000 | 11.085787 |
5 | 0.000103 | 0.001784 | 11.085787 |
10 | 0.000202 | 0.003501 | 11.085787 |
15 | 0.000280 | 0.005152 | 11.085787 |
20 | 0.000348 | 0.006745 | 11.085787 |
Dark soliton (Scheme 2)
t | \(\boldsymbol{L_{\infty}}\) | \(\boldsymbol{L_{2}}\) | \(\boldsymbol{I_{11}}\) ( 47 ) |
---|---|---|---|
0 | 0.000000 | 0.000000 | 11.085787 |
5 | 0.000103 | 0.001785 | 11.085787 |
10 | 0.000202 | 0.003502 | 11.085787 |
15 | 0.000280 | 0.005153 | 11.085787 |
20 | 0.000348 | 0.006746 | 11.085787 |
Dark soliton (Scheme 3)
t | \(\boldsymbol{L_{\infty}}\) | \(\boldsymbol{L_{2}}\) | \(\boldsymbol{I_{11}}\) ( 47 ) |
---|---|---|---|
0 | 0.000000 | 0.000000 | 11.085787 |
5 | 0.000103 | 0.001784 | 11.085787 |
10 | 0.000202 | 0.003501 | 11.085787 |
15 | 0.000280 | 0.005152 | 11.085787 |
20 | 0.000348 | 0.006745 | 11.085787 |
6.3 Interaction of two bright solitons
Interaction of two bright solitons for CNLS
t | \(\boldsymbol{I_{1}}\) | \(\boldsymbol{I_{2}}\) | \(\boldsymbol{I_{3}}\) | \(\boldsymbol{I_{4}}\) |
---|---|---|---|---|
0 | 5.164771 | 4.618162 | 10.169401 | 10.881909 |
10 | 5.164771 | 4.637349 | 10.198182 | 10.913261 |
40 | 5.164771 | 4.618895 | 10.171440 | 10.832564 |
80 | 5.164771 | 4.509860 | 10.007217 | 10.720618 |
100 | 5.164770 | 4.630145 | 10.186358 | 10.913433 |
120 | 5.164770 | 4.511656 | 9.988027 | 10.704388 |
140 | 5.164770 | 4.566811 | 10.079556 | 10.796694 |
150 | 5.164769 | 4.574081 | 10.092399 | 10.811464 |
7 Conclusion
In this work we have solved the chiral nonlinear Schrödinger equation numerically by deriving three different finite difference schemes. In Scheme 1, we derived a nonlinear implicit scheme, we have used a fixed point iterative method to solve the nonlinear block tridiagonal system obtained. In Schemes 2 and 3, we have derived two linearly implicit finite difference schemes. Crout’s method is used to solve the resulting linear block tridiagonal system. All numerical schemes we have derived in this work conserve the energy, and this indicates that no blow-up is expected during the simulation, and hence all schemes are stable. Concerning the accuracy, the proposed schemes all are of second order accuracy in both time and space directions. The linearized schemes, Scheme 2 and Scheme 3, are more efficient than Scheme 1 regarding the issue of the cpu execution time required.
Declarations
Acknowledgements
The authors thank the reviewers for their valuable suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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