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 transformationsMSC
35Q55 35Q60 35D991 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
References
- Bradley, CC, Sackett, CA, Tollett, JJ, Hulet, RG: Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions. Phys. Rev. Lett. 27, 1687-1690 (1995) View ArticleGoogle Scholar
- Saito, H, Ueda, M: Dynamically stabilized bright solitons in a two-dimensional Bose-Einstein condensate. Phys. Rev. Lett. 90, 040403 (2003) View ArticleGoogle Scholar
- Wang, DS, Hu, XH, Hu, J, Liu, WM: Quantized quasi-two-dimensional Bose-Einstein condensates with spatially modulated nonlinearity. Phys. Rev. A 81, 025604 (2010) View ArticleGoogle Scholar
- Porsezian, K, Hasegawa, A, Serkin, VN, Belyaeva, TL, Ganapathy, R: Dispersion and nonlinear management for femtosecond optical solitons. Phys. Lett. A 361, 504-508 (2007) View ArticleGoogle Scholar
- Calvo, GF, Belmonte-Beitia, J, Pérez-García, VM: Exact bright and dark spatial soliton solutions in saturable nonlinear media. Chaos Solitons Fractals 41, 1791-1798 (2009) MATHView ArticleGoogle Scholar
- Rose, HA, Weinstein, MI: On the bound states of the nonlinear Schrödinger equation with a linear potential. Physica D 30, 207-218 (1988) MATHMathSciNetView ArticleGoogle Scholar
- Oelza, D, Trabelsi, S: Analysis of a relaxation scheme for a nonlinear Schrödinger equation occurring in plasma physics. Math. Model. Anal. 19, 257-274 (2014) MathSciNetView ArticleGoogle Scholar
- Nore, C, Brachet, ME, Fauve, S: Numerical study of hydrodynamics using the nonlinear Schrödinger equation. Physica D 65, 154-162 (1993) MATHMathSciNetGoogle Scholar
- Wang, DS, Hu, XH, Liu, WM: Localized nonlinear matter waves in two-component Bose-Einstein condensates with time-and space-modulated nonlinearities. Phys. Rev. A 82, 023612 (2010) View ArticleGoogle Scholar
- Yan, ZY: Exact analytical solutions for the generalized non-integrable nonlinear Schrödinger equation with varying coefficients. Phys. Lett. A 374, 4838-4843 (2010) MATHMathSciNetView ArticleGoogle Scholar
- Wang, DS, Song, SW, Xiong, B, Liu, WM: Vortex states in rotating Bose-Einstein condensate with spatiotemporally modulated interaction. Phys. Rev. A 84, 053607 (2011) View ArticleGoogle Scholar
- Ablowitz, MJ, Segur, H: Soliton and the Inverse Scattering Transformation. SIAM, Philadelphia (1981) View ArticleGoogle Scholar
- Rogers, C, Schief, WK: Backlund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory. Cambridge University Press, Cambridge (2002) View ArticleGoogle Scholar
- Wang, DS, Wei, XQ: Integrability and exact solutions of a two-component Korteweg de Vries system. Appl. Math. Lett. 51, 60-67 (2016) MathSciNetView ArticleGoogle Scholar
- Matveev, VB, Salle, MA: Darboux Transformation and Solitons. Springer, Berlin (1991) View ArticleGoogle Scholar
- Hirota, R: Exact solution of the Korteweg de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192-1194 (1971) MATHView ArticleGoogle Scholar
- Ma, WX: Generalized bilinear differential equations. Stud. Nonlinear Sci. 2, 140-144 (2011) Google Scholar
- Belmonte-Beitia, J, Pérez-García, VM, Vekslerchik, V: Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities. Phys. Rev. Lett. 98, 064102 (2007) View ArticleGoogle Scholar
- Ma, WX, Chen, M: Direct search for exact solutions to the nonlinear Schrödinger equation. Appl. Math. Comput. 215, 2835-2842 (2009) MATHMathSciNetView ArticleGoogle Scholar
- Wang, DS, Zhang, DJ, Yang, J: Integrable properties of the general coupled nonlinear Schrödinger equations. J. Math. Phys. 51, 023510 (2010) MathSciNetView ArticleGoogle Scholar
- Yan, ZY: Envelope compactons and solitary patterns. Phys. Lett. A 355, 212-215 (2006) View ArticleGoogle Scholar
- Belmonte-Beitia, J, Pérez-García, VM, Vekslerchik, V, Konotop, VV: Localized nonlinear waves in systems with time- and space-modulated nonlinearities. Phys. Rev. Lett. 100, 164102 (2008) View ArticleGoogle Scholar
- Wang, DS, Shi, YR, Chow, KW, Yu, ZX, Li, XG: Matter-wave solitons in a spin-1 Bose-Einstein condensate with time-modulated external potential and scattering lengths. Eur. Phys. J. D 67, 242 (2013) View ArticleGoogle Scholar
- Wang, DS, Ma, YQ, Li, XG: Prolongation structures and matter-wave solitons in \(F=1\) spinor Bose-Einstein condensate with time-dependent atomic scattering lengths in an expulsive harmonic potential. Commun. Nonlinear Sci. Numer. Simul. 19, 3556-3569 (2014) MathSciNetView ArticleGoogle Scholar
- Ma, WX, Zhu, ZN: Solving the \((3 + 1)\)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm. Appl. Math. Comput. 218, 11871-11879 (2012) MATHMathSciNetView ArticleGoogle Scholar
- Ma, WX, Wu, HY, He, JS: Partial differential equations possessing Frobenius integrable decompositions. Phys. Lett. A 364, 29-32 (2007) MATHMathSciNetView ArticleGoogle Scholar
- Yan, ZY: The new tri-function method to multiple exact solutions of nonlinear wave equations. Phys. Scr. 78, 035001 (2008) MathSciNetView ArticleGoogle Scholar
- Wang, XL, Wu, ZH: New exact solutions and dynamics in \((3+1)\)-dimensional Gross-Pitaevskii equation with repulsive harmonic potential. Commun. Theor. Phys. 61, 583-589 (2014) View ArticleGoogle Scholar
- Ma, WX, Fuchssteiner, B: Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation. Int. J. Non-Linear Mech. 31, 329-338 (1996) MATHMathSciNetView ArticleGoogle Scholar
- Ma, WX: Trilinear equations, Bell polynomials, and resonant solutions. Front. Math. China 8, 1139-1156 (2013) MATHMathSciNetView ArticleGoogle Scholar