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Exact spatiotemporal soliton solutions to the generalized three-dimensional nonlinear Schrödinger equation in optical fiber communication
- Xiaoli Wang^{1}Email authorView ORCID ID profile and
- Jie Yang^{2}
https://doi.org/10.1186/s13662-015-0683-4
© Wang and Yang 2015
- Received: 16 September 2015
- Accepted: 2 November 2015
- Published: 10 November 2015
Abstract
In this paper, the exact spatiotemporal soliton solutions of the generalized \((3+1)\)-dimensional nonlinear Schrödinger equation with varying coefficients in optical fiber communication are obtained explicitly by using the similarity transformation. In addition, the propagation characteristics of the spatiotemporal optical solitons which can be dramatically affected by the complicated group velocity dispersion and self-phase modulation are discussed in detail.
Keywords
- nonlinear Schrödinger equation
- solitary wave
- spatiotemporal soliton
- similarity transformations
MSC
- 35Q55
- 35Q60
- 35D99
1 Introduction
The nonlinear Schrödinger (NLS) equation is one of the important mathematical models in many fields of physics, which has been widely applied in Bose-Einstein condensates [1–3], nonlinear optical fiber communication [4, 5], plasma physics [6, 7], hydrodynamics [8], and so on. Recently more and more people have been devoted to solving the exact solutions of the generalized NLS models [9–11]. Today, the temporal optical solitons of the NLS equation have been the objects of theoretical and experimental studies in optical fiber communication, and optical solitons are regarded as an important alternative to the next generation of ultrafast optical telecommunication systems. The study of optical solitons has reached the stage of a real-life application. The propagation of optical pulse in monomode optical fiber is governed by the NLS equation.
The generalized \((3+1)\)-dimensional nonlinear Schrödinger equation Eq. (1) is of considerable importance in optical fiber communication as it describes the amplification or absorption of pulses propagating in a monomode optical fiber with distributed dispersion and nonlinearity. In practical applications, the model is of primary interest not only for the amplification and compression of optical solitons in inhomogeneous systems, but also for the stable transmission of soliton control.
The paper is organized as follows. In Section 2, the similarity transformation is described, which converts a partial differential equation with variable coefficients into a family of first-order ordinary differential equations. In Section 3, several periodic traveling wave solutions are derived and some examples, which demonstrate the propagation characteristics of the spatiotemporal solitons, are given. Finally, a short conclusion is presented.
2 Similarity transformation
Jacobi elliptic functions
Solution | \(\boldsymbol{q_{2}}\) | \(\boldsymbol{q_{4}}\) | F |
---|---|---|---|
1 | \(-(1+M^{2})\) | \(M^{2}\) | sn |
2 | \(2M^{2}-1\) | \(-M^{2}\) | cn |
3 | \(2-M^{2}\) | −1 | dn |
4 | \(-(1+M^{2})\) | 1 | ns |
5 | \(2M^{2}-1\) | \(1-M^{2}\) | nc |
6 | \(2-M^{2}\) | \(M^{2}-1\) | nd |
7 | \(2-M^{2}\) | \(1-M^{2}\) | sc |
8 | \(2M^{2}-1\) | \(-M^{2}(1-M^{2})\) | sd |
9 | \(2-M^{2}\) | 1 | cs |
10 | \(-(1+M^{2})\) | \(M^{2}\) | cd |
11 | \(2M^{2}-1\) | 1 | ds |
12 | \(-(1+M^{2})\) | 1 | dc |
13 | \(M^{2}/2-1\) | \(M^{4}/4\) | sn/(1 + dn) |
14 | \(M^{2}/2-1\) | \(M^{4}/4\) | \(\mathrm{cn}/(\sqrt{1-M^{2}}+\mathrm{dn})\) |
By solving Eqs. (16)-(22) self-consistently, one obtains a set of conditions on the coefficients and parameters, necessary for Eq. (1) to have exact periodic wave solutions.
3 Analytical solutions and the propagation characteristics of the spatiotemporal solitons
As long as one chooses the constants according to the relations listed in Table 1 and substitutes the appropriate \(F(\xi)\) into Eq. (37), one obtains the exact periodic traveling wave solutions to the generalized \((3+1)\)-dimensional NLSE.
As an example, we select the solutions 1, 2, 13, 14 in Table 1 and the parameters \(b_{j}^{0}=k_{j}^{0}=f_{0}=1\) (\(j=1,2,3\)), \(\omega_{0}=c_{0}=0\), \(\gamma=\cos (z)\). According to the value of parameter \(h_{j}^{0}\), we list two classes.
3.1 The parameter \(h_{j}^{0}=1\) (\(j=1,2,3\))
Family 1
Family 2
Family 3
Family 4
3.2 The parameter \(h_{j}^{0}=0\) (\(j=1,2,3\))
Family 5
Taking parameters \(m=0.03\), \(s_{j}=0.0001\), we can obtain the periodic wave solution as follows.
Family 6
Taking parameters \(m=-0.03\), \(s_{j}=0.0001\), we can obtain the periodic wave solution as follows.
Family 7
Family 8
The evolution plots of solutions u in Families 5-8 are very close to the evolution plots in [28], and we omit the corresponding discussion for the limit of the length.
4 Conclusions
In conclusion, we have used similarity transformation to construct analytical spatiotemporal soliton solutions of the generalized \((3+1)\)-dimensional NLS equation with varying coefficients, and also investigated the propagation characteristics of the spatiotemporal solitons which can be dramatically affected by the complicated potential.
Declarations
Acknowledgements
This work is supported by the Science and Technology Project of Beijing Municipal Commission of Education (Grant No. KM201311232021).
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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