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Solitons for the cubic-quintic nonlinear Schrödinger equation with Raman effect in nonlinear optics

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Abstract

Considering the ultrashort optical soliton propagation in the non-Kerr media, the cubic-quintic nonlinear Schrödinger equation with Raman effect is studied through the dependent variable transformation and Hirota method. Based on symbolic computation, the bilinear form, the explicit one- and two-soliton solutions for the equation are presented. The constraint parametric condition for the existence of soliton solutions is also derived. Propagation characteristics and interaction behaviors of the solitons are graphically shown and discussed: (1) Overtaking elastic interactions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) propagation in parallel of the two solitons.

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References

  • Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, New York (1991)

    Book  MATH  Google Scholar 

  • Agrawal, G.P.: Nonlinear Fiber Optics, 4th edn. Academic Press, San Diego (2007)

    Google Scholar 

  • Alka Goyal, A., Gupta, R., Kumar, C.N., Raju, T.S.: Chirped femtosecond solitons and double-kink solitons in the cubic-quintic nonlinear Schrödinger equation with self-steepening and self-frequency shift. Phys. Rev. A 84, 0638309-1–0638309-6 (2011)

    Google Scholar 

  • Calogero, F., Eckhaus, W.: Nonlinear evolution equations, rescalings, model PDES and their integrability: I. Inverse Probl. 3, 229–262 (1987)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Chegini, N.G., Salaripanah, A., Mokhtari, R., Isvand, D.: Numerical solution of the regularized long wave equation using nonpolynomial splines. Nonlinear Dyn. 69, 459–471 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  • Clarkson, P.A., Tuszynski, J.A.: Exact solutions of the multidimensional derivative nonlinear Schrödinger equation for many-body systems near criticality. J. Phys. A 23, 4269–4288 (1990)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Fanjoux, G., Michaud, J., Delqué, M., Furfaro, L., Maillotte, H., Sylvestre, T.: Spatio-temporal dynamics of multicolor spatial Kerr solitons. Opt. Quantum Electron. 40, 271–279 (2008)

    Article  Google Scholar 

  • Gedalin, M., Scott, T.C., Band, Y.B.: Optical solitary waves in the higher order nonlinear Schrödinger equation. Phys. Rev. Lett 78, 448–451 (1997)

    Article  ADS  Google Scholar 

  • Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons. Phys. Rev. Lett. 27, 1192–1194 (1971)

    Article  ADS  MATH  Google Scholar 

  • Johnson, R.S.: On the modulation of water waves in the neighbourhood of kh\(\approx \)1.363. Proc. R. Soc. Lond. A 357, 131–141 (1977)

    Article  ADS  MATH  Google Scholar 

  • Kim, W.S., Moon, H.T.: Soliton-kink interactions in a generalized nonlinear Schrödinger system. Phys. Lett. A 266, 364–369 (2000)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  • Kodama, Y.: Optical solitons in a monomode fiber. J. Stat. Phys. 39, 597–614 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  • Kundu, A.: Landau-Lifeshitz and higher-order nonlinear systems gauge generated from nonlinear Schrödinger equation-type equations. J. Math. Phys. 25, 3433–3438 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  • Liu, W.J., Tian, B.: Symbolic computation on soliton solutions for variable-coefficient nonlinear Schrödinger equations in nonlinear optics. Opt. Quantum Electron. 43, 147–162 (2012)

    Article  MathSciNet  Google Scholar 

  • Lü, X., Peng, M.S.: Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model. Chaos 23, 013122-1–013122-7 (2013)

    ADS  Google Scholar 

  • Menezes, J.W.M., Sousa, J.R.R., Fraga, W.B., Lopes, V.C.M., Lima, F.T., Sobrinho, C.S., Sombra, A.S.B.: Spatiotemporal optical solitons in planar waveguide with periodically modulated cubic-quintic nonlinearity. Opt. Quantum Electron. 42, 179–192 (2010)

    Article  Google Scholar 

  • Mitschkem, F.M., Mollenauer, L.F.: Discovery of the soliton self-frequency shift. Opt. Lett. 11, 659–661 (1986)

    Article  ADS  Google Scholar 

  • Pushkarov, D.I., Tanev, S.: Bright and dark solitary wave propagation and bistability in the anomalous dispersion region of optical waveguides with third- and fifth-order nonlinearities. Opt. Commun. 124, 354–364 (1996)

    Article  ADS  Google Scholar 

  • Qi, F.H., Tian, B., Lü, X., Guo, R., Xue, Y.S.: Darboux transformation and soliton solutions for the coupled cubic-quintic nonlinear Schrödinger equations in nonlinear optics. Commun. Nonlinear. Sci. Numer. Simul. 17, 2372–2381 (2011)

    Article  ADS  Google Scholar 

  • Radhakrishnan, R., Kundu, A., Lakshmanan, M.: Coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity: Integrability and soliton interaction in non-Kerr media. Phy. Rev. E 60, 3314–3323 (1999)

    Article  ADS  Google Scholar 

  • Sarma, A.K.: Solitary wave solutions of higher-order NLSE with Raman and self-steepening effect in a cubic-quintic-septic medium. Commun. Nonlinear Sci. Numer. Simul. 14, 3215–3219 (2009)

    Article  ADS  Google Scholar 

  • Skarka, V., Berezhiani, V.I., Miklaszewski, R.: Spatiotemporal soliton propagation in saturating nonlinear optical media. Phys. Rev. E 56, 1080–1087 (1997)

    Article  ADS  Google Scholar 

  • Soto-Crespo, J.M., Pesquera, L.: Analytical approximation of the soliton solutions of the quintic complex Ginzburg-Landau equation. Phys. Rev. E. 56, 7288–7293 (1997)

    Article  ADS  Google Scholar 

  • Sun, Z.Y., Gao, Y.T., Yu, X., Liu, W.J., Liu, Y.: Bound vector solitons and soliton complexes for the coupled nonlinear Schrödinger equations. Phys. Rev. E 80, 066608-1–066608-11 (2009)

    ADS  Google Scholar 

  • Tian, B., Gao, Y.T.: Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: New transformation with burstons, brightons and symbolic computation. Phys. Lett. A 359, 241–248 (2006)

    Article  ADS  Google Scholar 

  • Wang, M.L., Zhang, J.L., Li, X.Z.: Solitary wave solutions of a generalized derivative nonlinear Schrödinger equation. Commun. Theor. Phys. 50, 39–42 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  • Wang, P., Tian, B.: Symbolic computation on the bright soliton solutions for the generalized coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity. Opt. Commun. 285, 3567–3577 (2012a)

    Article  ADS  MathSciNet  Google Scholar 

  • Wang, P., Tian, B.: Symbolic computation on soliton dynamics and Bäcklund transformation for the generalized coupled nonlinear Schrödinger equations with cubic-quintic nonlinearity. J. Mod. Opt. 59, 1786–1796 (2012b)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work has been supported by Nature Science Basic Research Plan in Shaanxi Province of China (Grant No. 2014JM8338), the China Postdoctoral Science Special Foundation (Grant No. 201104659), the China Postdoctoral Science Foundation (Grant No. 20100481322), the Foundation of State Key Lab on Integrated Service Networks (Grant No. ISN1003006), and the Fundamental Research Funds for the Central Universities (Grant No. K50511010023). This work is also part supported by 111 Project of China (B08038).

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Authors and Affiliations

  1. State Key Laboratory of Integrated Service Networks, School of Telecommunications Engineering, Xidian University, Xi’an , 710071, China

    Ping Wang, Tao Shang & Li Feng

  2. School of Science, Xidian University, Xi’an , 710071, China

    Ping Wang

  3. Department of Physics, Northwest University, Xi’an , 710069, China

    Yingjie Du

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  1. Ping Wang
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  2. Tao Shang
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  3. Li Feng
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  4. Yingjie Du
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Correspondence to Ping Wang.

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Wang, P., Shang, T., Feng, L. et al. Solitons for the cubic-quintic nonlinear Schrödinger equation with Raman effect in nonlinear optics. Opt Quant Electron 46, 1117–1126 (2014). https://doi.org/10.1007/s11082-013-9840-8

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  • DOI: https://doi.org/10.1007/s11082-013-9840-8

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