New spectral collocation algorithms for one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity
- Ali H Bhrawy^{1}Email author,
- Fouad Mallawi^{2} and
- Mohamed A Abdelkawy^{1, 3}
https://doi.org/10.1186/s13662-016-0752-3
© Bhrawy et al. 2016
Received: 28 October 2015
Accepted: 11 January 2016
Published: 25 January 2016
Abstract
A shifted Jacobi collocation method in two stages is constructed and used to numerically solve nonlinear Schrödinger equations (NLSEs) with a Kerr law nonlinearity, subject to initial-boundary conditions. An expansion in a series of spatial shifted Jacobi polynomials with temporal coefficients for the approximate solution is considered. The first stage, collocation at the shifted Jacobi Gauss-Lobatto (SJ-GL) nodes, is applied for a spatial discretization; its spatial derivatives occur in the NLSE with a treatment of the boundary conditions. This in all will produce a system of ordinary differential equations (SODEs) for the coefficients. The second stage is to collocate at the shifted Jacobi Gauss-Radau (SJ-GR-C) nodes in the temporal discretization to reduce the SODEs to a system of algebraic equations which is solved by an iterative method. Both stages can be extended to solve the two-dimensional NLSEs. Numerical examples are carried out to confirm the spectral accuracy and the efficiency of the proposed algorithms.
Keywords
1 Introduction
Spectral methods (see [1–8]) are accurate and efficient demanding less computations when solving an ordinary differential equation (ODE) or a partial differential equation (PDE) on a simple domain with smooth functions defined. The basic idea of the spectral methods is to express the approximate solution of the problem as a finite sum of certain basis functions (orthogonal polynomials or a combination of them) and then choose the coefficients in order to minimize the residual. The spectral collocation method is one type of spectral methods, which is more applicable and widely used to solve almost all linear and nonlinear differential equations [5–17].
1. Kerr law \(\mathbf{F}(s)=s\):
In particular, equation (1) with \(\mathbf{F}(s)=s\), (Kerr law case) appears in nonlinear optics in the context of soliton propagation through optical fibers. The first term represents the temporal evolution, the second term accounts for the dispersive effect of the solitons, the coefficient of γ represents a Kerr law nonlinearity and, finally, δ represents the coefficient of the driven term.
2. Power law \(\mathbf{F}(s)=s^{n}\):
Power law nonlinearity (a generalization of Kerr law nonlinearity) is exhibited in various fields, including semiconductors, photon processes, nonlinear plasmas, and weak turbulence [22].
3. Parabolic law \(\mathbf{F}(s)=s-k_{1} s^{2}\), where \(k_{1}\) is a constant:
Parabolic law (cubic-quintic nonlinearity) arises in the nonlinear p-toluene sulfonate crystals, interaction between Langmuir waves and electrons, the nonlinear interaction between the high-frequency Langmuir waves and the ion-acoustic waves.
4. Dual-power law \(\mathbf{F}(s)=s^{n}-k_{2} s^{2n}\), where \(k_{2} \) is a constant:
This law is used to model the saturation of the nonlinear refractive index, spatial solitons in photovoltaic-photorefractive materials, and organic and polymer materials [23].
5. Saturable law \(\mathbf{F}(s)=\frac{\lambda s}{1+\lambda s}\):
The variation of the dielectric constant of gas vapors while a laser beam propagates [24] can be accurately described by this law with \(\lambda> 0\). Also, optical nonlinearity saturates at a finite value of optical intensity in most materials and the soliton propagation in semiconductor-doped fibers can be modeled using the above form. The above form is observed in semiconductor-doped glass and other composite materials.
6. Log law \(\mathbf{F}(s)=\ln(s)\):
The Log-law NLSEs has been employed to model nonlinear behavior in several distinct scenarios in physics and in other areas of nonlinear science for instance, in nuclear physics [25], dissipative systems [26], capillary fluids [27], optics [28, 29], and magma transport [27].
Recently, the analytical and numerical solutions of different types of the previous classical Schrödinger equations were discussed in [30, 31], and for recent schemes for solving PDEs see [32–36]. Here, we focus on the application of shifted Jacobi Gauss-Lobatto collocation (SJ-GL-C) and Jacobi Gauss-Radau collocation (SJ-GR-C) methods in two consecutive stages for providing a high accurate numerical solution of the NLSEs with kerr-law nonlinearity. The proposed collocation scheme is applied for both temporal and spatial discretizations. First of all, the SJ-GL-C is used with a treatment of the boundary conditions for spatial discretization. Therefore, the NLSE with its boundary conditions is reduced to SODEs subject to a vector of initial values. Second, the SJ-GR-C is then applied for temporal discretization, which is more reasonable for solving initial value problems. Thereby, the problem is reduced to a system of algebraic equations which is easier to solve. In addition, this algorithm is developed to numerically solve the two-dimensional NLSEs. Finally, several numerical examples with comparisons showing the high accuracy and effectiveness of the proposed algorithm are presented.
This paper is organized as follows. We present a few preliminaries and some facts about shifted Jacobi polynomials in Section 2. Section 3 presents the collocation method for the one-dimensional NLSE subject to initial-boundary conditions. In Section 4 we address an extension to solve the two-dimensional version of NLSE. In Section 5, we propose the SJ-GR-C scheme to solve SODEs. Section 6 is devoted to solving four test problems. Finally, some concluding remarks are given in the last section.
2 Properties of shifted Jacobi polynomials
3 One-dimensional NLSE
In this section, the numerical algorithm is based on SJ-GL-C method to numerically solve NLSEs with initial-boundary conditions. The collocation points are selected at the SJ-GL interpolation nodes. The method is to discretize the NLSE in the spatial direction along with a new treatment for the conditions the system is subjected to, to create a SODEs of the unknown coefficients of the spectral expansion in the time direction.
4 Two-dimensional NLSE
5 System of differential equations
6 Numerical simulation and comparisons
This section reports several numerical examples to demonstrate the high accuracy and applicability of the proposed methods for solving one- and two-dimensional Schrödinger equations with a Kerr law nonlinearity. We also compare the results given from our scheme and those reported in the literature such as the Sinc-collocation and Sinc-Galerkin methods [39]. The comparisons reveal that our methods are very effective and convenient.
Example 1
Maximum absolute errors of problem ( 63 )
N = M | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{0}\) | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\frac{1}{2}}\) | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\frac{1}{2}}\) , \(\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{-\frac{1}{2}}\) | |||
---|---|---|---|---|---|---|
\(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | \(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | \(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | |
4 | 1.54 × 10^{−2} | 1.98 × 10^{−2} | 2.33 × 10^{−2} | 2.92 × 10^{−2} | 2.29 × 10^{−2} | 2.88 × 10^{−2} |
8 | 8.04 × 10^{−7} | 1.07 × 10^{−6} | 1.50 × 10^{−6} | 1.98 × 10^{−6} | 1.50 × 10^{−6} | 2.01 × 10^{−6} |
12 | 5.15 × 10^{−12} | 6.88 × 10^{−12} | 1.12 × 10^{−11} | 1.51 × 10^{−11} | 1.13 × 10^{−11} | 1.46 × 10^{−11} |
16 | 8.55 × 10^{−15} | 9.33 × 10^{−15} | 4.61 × 10^{−15} | 4.33 × 10^{−15} | 5.44 × 10^{−15} | 5.77 × 10^{−15} |
20 | 6.66 × 10^{−15} | 7.55 × 10^{−15} | 3.11 × 10^{−15} | 3.33 × 10^{−15} | 3.11 × 10^{−15} | 3.00 × 10^{−15} |
Example 2
Maximum absolute errors of problem ( 65 )
N = M | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{-\frac{1}{2}}\) | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\frac{1}{2}}\) | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\frac{1}{2}}\) , \(\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{-\frac{1}{2}}\) | |||
---|---|---|---|---|---|---|
\(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | \(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | \(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | |
4 | 2.20 × 10^{−3} | 2.64 × 10^{−3} | 2.14 × 10^{−3} | 2.43 × 10^{−3} | 2.13 × 10^{−2} | 2.46 × 10^{−3} |
8 | 7.64 × 10^{−8} | 7.48 × 10^{−8} | 1.29 × 10^{−7} | 1.24 × 10^{−7} | 4.33 × 10^{−8} | 3.87 × 10^{−8} |
12 | 3.81 × 10^{−13} | 3.77 × 10^{−13} | 9.46 × 10^{−13} | 9.46 × 10^{−13} | 4.32 × 10^{−13} | 4.40 × 10^{−13} |
16 | 5.50 × 10^{−15} | 4.101 × 10^{−15} | 3.69 × 10^{−15} | 3.44 × 10^{−15} | 2.33 × 10^{−15} | 2.33 × 10^{−15} |
20 | 3.66 × 10^{−15} | 3.66 × 10^{−15} | 9.83 × 10^{−15} | 7.22 × 10^{−15} | 2.33 × 10^{−15} | 2.33 × 10^{−15} |
Example 3
Maximum absolute errors of problem ( 67 )
N = M = K | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{0}\) | \(\boldsymbol{\alpha }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\frac{1}{2}}\) , \(\boldsymbol{\alpha }_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta }_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{-\frac{1}{2}}\) | ||
---|---|---|---|---|
\(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | \(\boldsymbol{M}_{\boldsymbol{1}}\) | \(\boldsymbol{M}_{\boldsymbol{2}}\) | |
4 | 1.29 × 10^{−3} | 1.30 × 10^{−3} | 2.12 × 10^{−3} | 2.21 × 10^{−3} |
6 | 3.05 × 10^{−6} | 3.04 × 10^{−6} | 5.52 × 10^{−6} | 5.61 × 10^{−6} |
8 | 3.78 × 10^{−9} | 3.73 × 10^{−9} | 7.43 × 10^{−9} | 7.48 × 10^{−9} |
10 | 6.43 × 10^{−11} | 6.68 × 10^{−11} | 3.30 × 10^{−10} | 4.84 × 10^{−10} |
Example 4
Maximum absolute errors with various choices of N , M , and K for problem ( 70 )
N = M = K | \(\boldsymbol{\alpha}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{0}\) | \(\boldsymbol{\alpha}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{-\frac{1}{2}}\) | ||||
---|---|---|---|---|---|---|
\(\boldsymbol{M}^{\boldsymbol{u}}_{\boldsymbol{E}}\) | \(\boldsymbol{M}^{\boldsymbol{v}}_{\boldsymbol{E}}\) | \(\boldsymbol{M}^{\boldsymbol{\psi}}_{\boldsymbol{E}}\) | \(\boldsymbol{M}^{\boldsymbol{u}}_{\boldsymbol{E}}\) | \(\boldsymbol{M}^{\boldsymbol{v}}_{\boldsymbol{E}}\) | \(\boldsymbol{M}^{\boldsymbol{\psi}}_{\boldsymbol{E}}\) | |
4 | 2.73 × 10^{−5} | 2.42 × 10^{−5} | 3.65 × 10^{−5} | 1.13 × 10^{−5} | 1.13 × 10^{−5} | 2.15 × 10^{−5} |
6 | 6.18 × 10^{−8} | 4.00 × 10^{−8} | 7.36 × 10^{−8} | 3.25 × 10^{−8} | 2.17 × 10^{−8} | 3.36 × 10^{−8} |
8 | 6.21 × 10^{−11} | 3.11 × 10^{−11} | 6.85 × 10^{−11} | 2.61 × 10^{−11} | 2.95 × 10^{−11} | 3.05 × 10^{−11} |
Comparison based on relative errors for problem ( 70 )
N = M = K | \(\boldsymbol{\alpha}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{0}\) | N = M = K | \(\boldsymbol{\alpha}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\frac{1}{2}}\) | ||||
---|---|---|---|---|---|---|---|
\(\boldsymbol{R}^{\boldsymbol{u}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{v}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{\psi}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{u}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{v}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{\psi}}_{\boldsymbol{M}}\) | ||
4 | 5.06 × 10^{−5} | 2.88 × 10^{−5} | 3.65 × 10^{−5} | 4 | 6.32 × 10^{−5} | 4.07 × 10^{−5} | 4.84 × 10^{−5} |
6 | 1.14 × 10^{−7} | 4.76 × 10^{−8} | 7.36 × 10^{−8} | 6 | 1.70 × 10^{−7} | 8.88 × 10^{−8} | 1.19 × 10^{−7} |
8 | 1.15 × 10^{−10} | 3.70 × 10^{−11} | 6.85 × 10^{−11} | 8 | 2.00 × 10^{−10} | 9.04 × 10^{−11} | 1.32 × 10^{−10} |
N = M = K | \(\boldsymbol{\alpha}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\alpha}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{1}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{2}}\boldsymbol{=}\boldsymbol{\beta}_{\boldsymbol{3}}\boldsymbol{=}\boldsymbol{-\frac{1}{2}}\) | ( N , M , K ) | Symmetric Sinc-Galerkin method [ 39 ] | ||||
---|---|---|---|---|---|---|---|
\(\boldsymbol{R}^{\boldsymbol{u}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{v}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{\psi}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{u}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{v}}_{\boldsymbol{M}}\) | \(\boldsymbol{R}^{\boldsymbol{\psi}}_{\boldsymbol{M}}\) | ||
4 | 3.38 × 10^{−5} | 1.34 × 10^{−5} | 2.15 × 10^{−5} | (8,8,1,000) | 1.56 × 10^{−5} | 1.40 × 10^{−5} | 2.30 × 10^{−5} |
6 | 6.03 × 10^{−8} | 2.41 × 10^{−8} | 3.36 × 10^{−8} | (16,16,1,000) | 4.51 × 10^{−7} | 5.04 × 10^{−7} | 6.77 × 10^{−7} |
8 | 4.84 × 10^{−11} | 3.51 × 10^{−11} | 3.05 × 10^{−11} | (32,32,1,000) | 2.98 × 10^{−9} | 4.90 × 10^{−9} | 5.62 × 10^{−9} |
7 Conclusions
In this paper, we have proposed a collocation algorithm to introduce an accurate numerical solution for the one-dimensional nonlinear NLSEs with initial-boundary conditions. The core of the proposed method was to discretize the NLSE in the spatial direction by the SJ-GL-C method, along with a new treatment for the subjected conditions, to create a system of SODEs of the unknown coefficients of the spectral expansion in the time direction. An efficient numerical integration process for SODEs was investigated based on the SJ-GR-C method. The proposed method was extended to solve the two-dimensional NLSEs. The main advantage of the proposed algorithm is that, on adding few terms of the SJ-GL-C and SJ-GR-C nodes, a good approximation of the exact solution of the problem was achieved. Comparisons between our approximate solutions of the problems with their exact solutions and with the approximate solutions achieved by other methods were introduced to confirm the validity and accuracy of our scheme.
Declarations
Acknowledgements
This article was funded by the Deanship of Scientific Research DSR, King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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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