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Elliptic function solutions, modulation instability and optical solitons analysis of the paraxial wave dynamical model with Kerr media

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Abstract

The optical solitary waves explain non-dispersive and non-diffractive spatiotemporal localized waves envelopes promulgating in the media of optical Kerr. These propagations are generally described by the nonlinear Schrö dinger equation. The modified extended mapping technique is utilized to assemble the solitons, solitary waves and rational solutions of time-dependent dimensionless paraxial wave equation. The obtained different sorts of wave solutions encompass key applications in physics and engineering. By giving appropriate values to parameter, special sorts of solitary waves configuration can be displayed graphically. The physical interpretation of the solution can be understood through the structure. The stability of this wave equation is investigated via using modulational instability analysis which authenticates that all soliton solutions are stable and exact. Several analytical results and working out have confirmed the strength and efficacy of the current technique.

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References

  • Agrawal, G.P.: Nonlinear fiber optics, 5th edn. Academic Press, New York (2013)

    MATH  Google Scholar 

  • Ahmed, I., Seadawy, A.R., Lu, D.: Mixed lump-solitons, periodic lump and breather soliton solutions for (2+ 1)-dimensional extended Kadomtsev–Petviashvili dynamical equation. Int. J. Mod. Phys. B 33(5), 1950019 (2019)

    Article  MathSciNet  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R.: Dianchen Lu, Modulation stability and optical soliton solutions of nonlinear Schrödinger equation with higher order dispersion and nonlinear terms and its applications. Superlattices Microstruct. 112, 422–434 (2017)

    Article  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D., Wang, J.: Travelling wave solutions of Drinfel’d–Sokolov-Wilson, Whitham–Broer–Kaup and (2+1)-dimensional Broer–Kaup–Kupershmit equations and their applications. Chin. J. Phys. 55, 780–797 (2017)

    Article  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Bright-dark solitary wave solutions of generalized higher-order nonlinear Schrödinger equation and its applications in optics. J. Electromagn. Waves Appl. 31, 1711–1721 (2017a)

    Article  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Exact bright-dark solitary wave solutions of the higher-order cubic-quintic nonlinear Schrodinger equation and its stability. Optik 128, 40–49 (2017b)

    Article  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D., Wang, J.: Optical soliton solutions of unstable nonlinear Schrödinger dynamical equation and stability analysis with applications. Optik 157, 597–605 (2018a)

    Article  ADS  Google Scholar 

  • Arshad, M., Seadawy, A.R., Lu, D.: Modulation stability and dispersive optical soliton solutions of higher order nonlinear Schrödinger equation and its applications in mono-mode optical fibers. Superlattices Microstruct. 113, 419–429 (2018b)

    Article  ADS  Google Scholar 

  • Baronio, F., De Angelis, C., Pioger, P.H., Couderc, V., Barthelemy, A.: Reflection of quadratic solitons at the boundary of nonlinear media. Opt. Lett. 29, 986–988 (2004)

    Article  ADS  Google Scholar 

  • Baronio, F., Wabnitz, S., Kodama, Y.: Optical Kerr spatiotemporal dark-lump dynamics of hydrodynamic origin. Phys. Rev. Lett. 116, 638–647 (2016)

    Article  Google Scholar 

  • Boyd, R.: Nonlinear Optics, 3rd edn. Academic Press, London (2008)

    Google Scholar 

  • Cheng-Lin, B., Hong, Z., Wei-Tao, W.: Exact solutions to extended nonlinear Schrodinger equation in monomode optical fiber. Commun. Theor. Phys. 45, 131–144 (2006)

    Article  ADS  Google Scholar 

  • Conti, C., Trillo, S., Di Trapani, P., Valiulis, G., Piskarskas, A., Jedrkiewicz, O., Trull, J.: Nonlinear Electromagnetic X waves. Phys. Rev. Lett. 90, 342–361 (2003)

    Article  Google Scholar 

  • Farah, N., Seadawy, A.R., Ahmad, S., Rizvi, S.T.R., Younis, M.: Interaction properties of soliton molecules and Painleve analysis for nano bioelectronics transmission model. Opt. Quantum Electron. 52, 1–15 (2020)

    Article  Google Scholar 

  • Jedrkiewicz, O., Picozzi, A., Clerici, M., Faccio, D., Di Trapani, P.: Emergence of X-shaped spatiotemporal coherence in optical waves. Phys. Rev. Lett. 97, 572–583 (2006)

    Article  Google Scholar 

  • Kalogiratou, Z., Monovasilis, T., Simos, T.E.: Numerical solution of the two dimensional time-independent Schrödinger equation with exponential-fitting methods. J. Math. Chem. 37(3), 271–279 (2005)

    Article  MathSciNet  Google Scholar 

  • Koprinkov, I.G., Suda, A., Wang, P., Midorikawa, K.: Self-compression of high-intensity femtosecond optical pulses and spatiotemporal soliton generation. Phys. Rev. Lett. 84, 38–47 (2000)

    Article  Google Scholar 

  • Kudryashov, N.A., Soukharev, M.B., Demina, M.V.: Elliptic traveling waves of the Olver equation. Commun. Nonlinear Sci. Numer. Simul. 17, 4104–4114 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  • Kumar, M., Kumar, R.: Soliton solutions of KD system using similarity transformations method. Comput. Math. Appl. 73, 701–712 (2017)

    Article  MathSciNet  Google Scholar 

  • Najafi, M., Arbabi, S.: Exact solutions of five complex nonlinear Schrödinger equations by semi-inverse variational principle. Commun. Theor. Phys. 62, 301–307 (2014)

    Article  MathSciNet  Google Scholar 

  • Nasreen, N., Lu, D., Arshad, M.: Optical soliton solutions of nonlinear Schrdinger equation with second order spatiotemporal dispersion and its modulation instability. Optik 161, 221–229 (2018)

    Article  ADS  Google Scholar 

  • Palacios, S.L., Fernndez-Diaz, J.M.: Bright solitary waves in high dispersive media with parabolic nonlinearity law: the influence of third order dispersion. J. Mod. Opt. 48, 1691–1699 (2001)

    Article  ADS  Google Scholar 

  • Pioger, P.H., Couderc, V., Lefort, L., Barthelemy, A., Baronio, F., De Angelis, C., Min, Y., Quiring, V., Sohler, W.: Spatial trapping of short pulses in Ti-indiffused LiNbO3 waveguides. Opt. Lett. 27, 2182–2184 (2002)

    Article  ADS  Google Scholar 

  • Potasek, M.J.: Modulation instability in an extended nonlinear Schrödinger equation. Opt. Lett. 12(11), 921–923 (1987)

    Article  ADS  Google Scholar 

  • Seadawy, A.R.: Approximation solutions of derivative nonlinear Schrödinger equation with computational applications by variational method. Eur. Phys. J. Plus 130, 182 (2015)

    Article  Google Scholar 

  • Seadawy, A.R.: Stability analysis solutions for nonlinear three-dimensional modified Korteweg-de Vries–Zakharov–Kuznetsov equation in a magnetized electron-positron plasma. Phys. A: Stat. Mech. Appl. Phys. A 455, 44–51 (2016)

    Article  MathSciNet  Google Scholar 

  • Seadawy, A.: Two-dimensional interaction of a shear flow with a free surface in a stratified fluid and its a solitary wave solutions via mathematical methods. Eur. Phys. J. Plus 132, 518 (2017)

    Article  Google Scholar 

  • Seadawy, A.R.: Solitary wave solutions of tow-dimensional nonlinear Kadomtsev–Petviashvili dynamic equation in a dust acoustic plasmas. Pramana J. Phys. 89(49), 1–11 (2017)

    ADS  Google Scholar 

  • Seadawy, A.: The generalized nonlinear higher order of KdV equations fromthe higher order nonlinear Schrodinger equation and its solutions. Optik - Int. J. Light Electron Opt. 139, 31–43 (2017)

    Article  Google Scholar 

  • Seadawy, A.R.: Travelling wave solutions of a weakly nonlinear two-dimensional higher order Kadomtsev–Petviashvili dynamical equation for dispersive shallow water waves. Eur. Phys. J. Plus 132(29), 1–13 (2017)

    Google Scholar 

  • Seadawy, A., El-Rashidy, K.: Dispersive Solitary wave solutions of Kadomtsev–Petviashivili and modified Kadomtsev–Petviashivili dynamical equations in unmagnetized dust plasma. Results Phys. 8, 1216–1222 (2018)

    Article  ADS  Google Scholar 

  • Seadawy, A.R., Arshad, M., Lu, D.: Stability analysis of new exact traveling-wave solutions of new coupled KdV and new coupled Zakharov–Kuznetsov systems. Eur. Phys. J. Plus 132, 162 (2017)

    Article  Google Scholar 

  • Seadawy, A.R., Arshad, M., Lu, D.: Modulation stability analysis and solitary wave solutions of nonlinear higher-order Schrodinger dynamical equation with second-order spatiotemporal dispersion. Indian J Phys (2018). https://doi.org/10.1007/s12648-018-01361-y

    Article  Google Scholar 

  • Seadawy, A.R., Lu, D., Arshad, M.: Stability analysis of solitary wave solutions for coupled and (2+1)-dimensional cubic Klein-Gordon equations and their applications. Commun. Theor. Phys. 69, 676–686 (2018)

    Article  MathSciNet  ADS  Google Scholar 

  • Seadawy, A.R., Iqbal, M., Lu, D., Seadawy, M., Iqbal, D.L.: Nonlinear wave solutions of the Kudryashov–Sinelshchikov dynamical equation in mixtures liquid-gas bubbles under the consideration of heat transfer and viscosity. J. Taibah Univ. Sci. 13(1), 1060–1072 (2019)

    Article  Google Scholar 

  • Triki, H., Biswas, A.: Dark solitons for a generalized nonlinear Schrodinger equation with parabolic law and dual power nonlinearity. Math. Methods Appl. Sci. 34, 958–962 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  • Weiner, A.: Ultrafast Optics. Wiley, New York (2009)

    Book  Google Scholar 

  • Zakharov, V.E.: Collapse of Langmuir waves. Sov. Phys. JETP 35, 908–914 (1972)

    ADS  Google Scholar 

  • Zhang, L.H., Si, J.G.: New soliton and periodic solutions of (1+2)-dimensional nonlinear Schrodinger equation with dual power nonlinearity. Commun. Nonlinear Sci. Numer. Simul. 15, 2747–2754 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  • Zhang, L., Lin, Y., Liu, Y.: New solitary wave solutions for two nonlinear evolution equations. Comput. Math. Appl. 67, 1595–1606 (2014)

    Article  MathSciNet  Google Scholar 

  • Zhao, Q., Wu, L.: Darboux transformation and explicit solutions to the generalized TD equation. Appl. Math. Lett. 67, 1–6 (2017)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the China Post-doctoral science foundation, Peoples Republic of China (PRC) (Grant Nos. 2019M651715).

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Authors and Affiliations

  1. Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu, 212013, People’s Republic of China

    Muhammad Arshad & Dianchen Lu

  2. Mathematics Department, Faculty of Science, Taibah University, Medina , Saudi Arabia

    Aly R. Seadawy

  3. Department of Mathematics, University of Okara, Okara, Pakistan

    Muhammad Shoaib Saleem

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  1. Muhammad Arshad
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  2. Aly R. Seadawy
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  3. Dianchen Lu
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  4. Muhammad Shoaib Saleem
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Correspondence to Aly R. Seadawy.

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Arshad, M., Seadawy, A.R., Lu, D. et al. Elliptic function solutions, modulation instability and optical solitons analysis of the paraxial wave dynamical model with Kerr media. Opt Quant Electron 53, 7 (2021). https://doi.org/10.1007/s11082-020-02637-6

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