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Diverse soliton structures for fractional nonlinear Schrodinger equation, KdV equation and WBBM equation adopting a new technique

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Abstract

Nonlinear fractional evolution equations are significant models for depicting intricate physical phenomena arise in nature. In this exploration, we concentrate to disentangle the space and time fractional nonlinear Schrodinger equation (NLSE), Korteweg-De Vries (KdV) equation and the Wazwaz-Benjamin-Bona-Mahony (WBBM) equation bearing the noteworthy significance in accordance to their respective position. A composite wave variable transformation is used to reduce the declared equations into ordinary differential equations. A successful implementation of a new technique called improved auxiliary equation technique collects enormous wave solutions which are physically appeared in diverse profiles such as kink, bell, cuspon, peakon, compacton and periodic etc. A comparable study of gained solutions with existing results in the literature ensures the diversity and novelty of this work. The improved auxiliary equation technique is appeared as efficient and concise tool which deserves further use to unravel any other nonlinear evolution equations arise in various physical sciences like applied mathematics, mathematical physics and engineering.

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References

  • Ablowitz, M.J., Clarkson, P.A.: Solitons Nonlinear Evolution Equations and Inverse Scattering Transform. pp. 1–480. Cambridge University Press, Cambridge, UK (1991) 

    Book  MATH  Google Scholar 

  • Aguilar, J.F.G., Saad, K.M., Baleanu, D.: Fractional dynamics of an erbium-doped fiber laser model. Opt. Quant. Elec. 51(9), 1–18 (2019)

    Google Scholar 

  • Akbar, M.A., Ali, N.H.M., Islam, M.T.: Multiple closed form solutions to some fractional order nonlinear evolution equations in physics and plasma physics. AIMS. Math. 4(3), 397–411 (2019a)

    Article  MathSciNet  Google Scholar 

  • Akbar, M.A., Ali, N.H.M., Tanjim, T.: Outset of multiple soliton solutions to the nonlinear Schrodinger equation and the coupled Burgers equations. J. Phys. Commun. 3(9), 1–11 (2019b)

    Article  Google Scholar 

  • Akram, U., Seadawy, A.R., Rizvi, S.T.R., Younis, M., Althobaiti, S., Sayed, S.: Traveling wave solutions for the fractional Wazwaz-Benjamin-Bona-Mahony model in arising shallow water waves. Res. Phys. 20, 103725 (2021)

    Google Scholar 

  • Alderemy, A.A., Attia, R.A.M., Alzaidi, J.F., Lu, D., Khater, M.M.A.: Analytical and semi-analytical wave solutions for longitudinal wave equation via modified auxiliary equation method and Adomian Decomposition method. Thermal Sci 23(6), 1943–1957 (2019)

    Article  Google Scholar 

  • Alqudah, M.A., Ravichandran, C., Abdeljawad, T., Valliammal, N.: New results on Caputo fractional-order neutral differential inclusions without compactness. Adv. Diff. Eq. 2019(1), 1–14 (2019)

    MathSciNet  Google Scholar 

  • Al-Qurashi, M.M., Yusuf, A., Aliyu, A.I., Inc, M.: Optical and other solitons for the fourth-order dispersive nonlinear Schrodinger equation with dual-power law nonlinearity. Superlattices Microstruct. 105, 183–197 (2017)

    Article  ADS  Google Scholar 

  • Aslan, E.C., Inc, M.: Soliton solutions of NLSE with quadratic-cubic nonlinearity and stability analysis. Waves Ran. Com. Med. 27(4), 594–601 (2017)

    Article  ADS  MathSciNet  Google Scholar 

  • Aslan, E.C., Tchier, F., Inc, M.: On optical solutions of the Schrodinger-Hirota equation with power law nonlinearity in optical fibers. Superlattices, Microstruc 105, 48–55 (2017)

    Article  ADS  Google Scholar 

  • Ates, E., Inc, M.: Travelling wave solutions of generalized Kliein-Gordon equations using Jacobi elliptic functions. Nonlinear Dyn. 88(3), 2281–2290 (2017)

    Article  MATH  Google Scholar 

  • Attia, R.A.M., Lu, D., Khater, M.M.A.: Structures of new solitary solutions for the Schwarzian Korteweg De Vries equation and (2+1)-Ablowitz-Kaup-Newell-Segur equation. Phys. J. 1(3), 234–254 (2018)

    Google Scholar 

  • Attia, R.A.M., Lu, D., Khater, M.M.A.: Chaos and relativistic energy-momentum of the nonlinear time fractional duffing equation. Math. Com. Appl. 24(1), 1–13 (2019)

    MathSciNet  Google Scholar 

  • Bekir, A., Guner, O.A.: Exact solutions of nonlinear fractional differential equations by (G’/G)-expansion method. Chin. Phys. B 22(11), 404–409 (2013)

    Article  Google Scholar 

  • Bibi, S., Mohyud-Din, S.T., Khan, U., Ahmed, N.: Khater method for nonlinear sharma tasso-olever (STO) equation of fractional order. Res. Phys. 7, 4440–4450 (2017)

    Google Scholar 

  • Cheema, N., Younis, M.: New and more general traveling wave solutions for nonlinear Schrodinger equation. Waves Ran. Com. Med. 26(1), 30–41 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Dascioglu, A., Culha, S., Bayram, D.V.: New analytical solutions of the space fractional KdV equation in terms of Jacobi elliptic functions. New Trends Math. Sci. 5(4), 232–241 (2017)

    Article  Google Scholar 

  • Durur, H., Ilhan, E., Bulut, H.: Novel complex wave solutions of the (2+1)-dimensional hyperbolic nonlinear Schrodinger equation. Fractal Fractional 4(3), 1–9 (2020)

    Article  Google Scholar 

  • Gao, W., Veeresha, P., Prakasha, D.G., Baskonus, H.M.: New numerical simulation for fractional Benney-lLin equation arising in falling film problems using two novel techniques. Numer. Meth. Partial Diff. Equ. 37(1), 210–243 (2021)

    Article  Google Scholar 

  • Ghanbari, B., Kumar, S., Kumar, R.: A study of behaviour for immune and tumor cells in immunogenetic tumour model with non-singular fractional derivative. Chaos, Solitons, Fractals 133, 1–14 (2020)

    Article  MathSciNet  Google Scholar 

  • Goufo, E.F., Kumar, S., Mugisha, S.B.: Similarities in a fifth-order evolution equation with and with no singular kernel. Chaos, Solitons, Fractals 130, 1–7 (2020)

    Article  MathSciNet  Google Scholar 

  • Hemida, K.M., Gepreel, K.A., Mohamed, M.S.: Analytical approximate solution to the time-space nonlinear partial fractional differential equations. Int. J. Pure Appl. Math. 78(2), 233–243 (2012)

    MATH  Google Scholar 

  • Inc, M.: New exact solutions for the ZK-MEW equation by using symbolic computation. Appl. Math. Com. 189(1), 508–513 (2007a)

    Article  MathSciNet  MATH  Google Scholar 

  • Inc, M.: New compacton and solitary pattern solutions of the nonlinear modified dispersive Klein-Gordon equations. Chaos, Solitons, Fractals 33(4), 1275–1284 (2007b)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Inc, M.: New type soliton solutions for the Zhiber-Shabat and related equations. Optik 138, 1–7 (2017)

    Article  ADS  Google Scholar 

  • Inc, M., Kilic, B.: Classification of traveling wave solutions for time-fractional fifth-order KdV-like equation. Waves Ran. Com. Med. 24(4), 393–403 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Inc, M., Kilic, B.: The first integral method for the perturbed Wadati-Segur-Ablowitz equation with time dependent coefficient. Kuwait J. Sci. 43(1), 84–94 (2016)

    MathSciNet  MATH  Google Scholar 

  • Inc, M., Ates, E., Tchier, F.: Optical solitons of the coupled nonlinear Schrodinger’s equation with Spatiotemporal dispersion. Nonlinear Dyn. 85(2), 1319–1329 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Islam, M.T., Akter, M.A.: Distinct solutions of nonlinear space-time fractional evolution equations appearing in mathematical physics via a new technique. Partial Diff. Equ. Appl. Math. 3, 1–10 (2021)

    Google Scholar 

  • Islam, M.T., Akbar, M.A., Azad, A.K.: A Rational -expansion method and its application to the modified KdV-Burgers equation and the (2+1)-dimensional Boussinesq equation. Non. Studies 6(4), 1–11 (2015)

    MathSciNet  MATH  Google Scholar 

  • Jassim, H.K., Baleanu, D.: A novel approach for Korteweg-de Vries equation of fractional order. J. Appl. Comput. Mech. 5(2), 192–198 (2019)

    Google Scholar 

  • Kaplan, M., Unsal, O., Bekir, A.: Exact solutions of nonlinear Schrodinger equation by using symbolic computation. Math. Meth. Appl. Sci. 39(8), 2093–2099 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  • Khader, M.M., Saad, K.M.: Numerical studies of the fractional Korteweg-de Vries, Korteweg-de Vries-Burgers’ and Burgers’ equations. Proc Natl Acad Sci India Sect A Phys Sci 91(1), 67–77 (2021)

    Article  MathSciNet  Google Scholar 

  • Khader, M.M., Saad, K.M., Hammouch, Z., Baleanu, D.: A spectral collocation method for solving fractional KdV and KdV-Burgers’ equations with non-singular kernel derivatives. Appl. Numer. Math. 161, 137–146 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Khalil, R., Horani, M.A., Yousef, A., Sababheh, M.A.M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Khater, M.M.A., Seadawy, A.R., Lu, D.: Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and time-fractional Cahn-Allen equation. Res. Phys. 7, 2325–2333 (2017)

    Google Scholar 

  • Khater, M.M.A., Seadawy, A.R., Lu, D.: New optical soliton solutions for nonlinear complex fractional Schrodinger equation via new auxiliary equation method and novel (G’/G)-expansion method. Pramana-J. Phys. 90(5), 1–20 (2018)

    Article  ADS  Google Scholar 

  • Khater, M.M.A., Attia, R.A.M., Lu, D.: Modified auxiliary equation method versus three nonlinear fractional biological models in present explicit wave solutions. Math. Com. Appl. 24(1), 1–8 (2019a)

    MathSciNet  Google Scholar 

  • Khater, M.M.A., Lu, D., Attia, R.A.M.: Dispersive long wave of nonlinear fractional Wu-Zhang system via a modified auxiliary equation method. AIP Adv. 9(2), 1–10 (2019b)

    Article  Google Scholar 

  • Khater, M.M.A., Mousa, A.A., El-Shorbagy, M.A., Attia, R.A.M.: Abundant novel wave solutions of nonlinear Klein-Gordon-Zakharov (KGZ) model. Eur. Phys. J. plus 136(5), 1–11 (2021)

    Article  Google Scholar 

  • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J. (2006) Theory and applications of fractional differential equations. North-Holland Math. pp. 204–400. Studies, Amsterdam, The Netherlands: Elsevier Sci

  • Kilic, B., Inc, M.: On optical solitons of the resonant Schrodinger’s equation in optical fibers with dual-power law nonlinearity and time-dependent. Waves Ran. Com. Med. 25(3), 334–341 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Kilic, B., Inc, M.: Soliton solutions for the Kundu-Eckhaus equation with the aid of unified algebraic and auxiliary equation expansion methods. J. Elec-Mag. Waves Appl. 30(7), 871–879 (2016)

    Article  Google Scholar 

  • Kumar, S.: A new analytical modelling for fractional telegraph equation via Laplace transform. Appl. Math. Mod. 38(13), 3154–3163 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, S., Kumar, R., Cattani, C., Samet, B.: Chaotic behaviour of fractional predator-prey dynamical system. Chaos, Solitons, Fractals 135, 1–11 (2020a)

    Article  MathSciNet  Google Scholar 

  • Kumar, S., Ahmadian, A., Kumar, R., Kumar, D., Singh, J., Baleanu, D., Salimi, M.: An efficient numerical method for fractional SIR epidemic model of infectious disease by using Bernstein wavelets. Math. 8(4), 1–9 (2020b)

    Google Scholar 

  • Kumar, S., Ghosh, S., Samet, B., Goufo, E.F.D.: An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator. Math. Meth. Appl. Sci. 43(9), 6062–6080 (2020c)

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, S., Kumar, R., Agarwal, R.P., Samet, B.: A study of fractional Lotka-Volterra population model using Haar wavelet and Adams-Bashforth-Moulton methods. Math. Meth. Appl. Sci. 43(8), 5564–5578 (2020d)

    Article  MathSciNet  MATH  Google Scholar 

  • Kumar, S., Chauhan, R.P., Momani, S., Hadid, S.: Numerical investigations on COVID-19 model through singular and non-singular fractional operators. Numer. Meth. Partial. Diff. Equ. (2020e)

  • Kumar, S., Kumar, R., Momani, S., Hadid, S.: A study on fractional COVID-19 disease model by using Hermite wavelets. Math. Meth. Appl. Sci. (2021)

  • Liu, J., Zhang, Y.: Analytical study of exact solutions of the nonlinear Korteweg-de Vries equation with space-time fractional derivatives. Mod. Phys. Lett. B 32(2), 1–8 (2018)

    Article  MathSciNet  Google Scholar 

  • Lu, D., Seadawy, A.R., Arshad, M.: Application of extended simple equation method on unstable Schrodinger equations. Optik 140, 136–144 (2017)

    Article  ADS  Google Scholar 

  • Ma, W.X., Lee, J.H.: A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo-Miwa equation. Chaos Solitons Fractals 42(3), 1356–1363 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Maarouf, N., Hilal, K.: Invariant analysis, analytical solutions, and conservation laws for two-dimensional time fractional Fokker-Planck equation. J. Fun. Spaces 2021, 1–9 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  • Martinez, H.Y., Aguilar, J.F.G., Atangana, A.: First integral method for nonlinear differential equations with conformable derivative. Math. Model. Nat. Phenom. 13(1), 1–11 (2018)

    MATH  Google Scholar 

  • Miller, K.S., Ross, B.: An introduction to the fractional calculus and fractional differential equations. pp. 1–300. John Wiley & Sons, New York, NY, USA (1993)

    MATH  Google Scholar 

  • Neirameh, A., Parvaneh, F.: Analytical solitons for the space-time conformable differential equations using two efficient techniques. Adv. Diff. Equ. 2021(1), 1–14 (2021)

    Article  MathSciNet  Google Scholar 

  • Odibat, Z.: A Riccati equation approach and travelling wave solutions for nonlinear evolution equations. Int. J. Appl. Comput. Math. 3(1), 1–13 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  • Oldham, K.B., Spanier, J.: The Fractional Calculus. pp. 1–300. Academic Press, New York, NY, USA (1974)

    MATH  Google Scholar 

  • Panda, S.K., Ravichandran, C., Hazarika, B.: Results on system of Atangana-Baleanu fractional order Willis aneurysm and nonlinear singularly perturbed boundary value problems. Chaos, Solitons, Fractals 142, 110390 (2021)

    Article  MathSciNet  Google Scholar 

  • Pandir, Y., Duzgun, H.H.: New exact solutions of the space-time fractional cubic Schrodinger equation using the new type F-expansion method. Waves Ran. Com. Med. 29(3), 425–434 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  • Podlubny, I. (1999) Fractional differential equations. Math. Sci. Eng. pp. 198–400. Academic Press, San Diego, Calif, USA

  • Ravichandran, C., Trujillo, J.J.: Controllability of impulsive fractional functional integro-differential equations in Banach spaces. J. Fun. Spaces Appl. 2013, 1–8 (2013)

    MathSciNet  MATH  Google Scholar 

  • Rezazadeh, H., Korkmaz, A., Eslami, M., Alizamini, S.M.M.: A large family of optical solutions to Kundu-Eckhaus model by a new auxiliary equation method. Opt. Quan. Elec. 51(3), 1–12 (2019)

    Google Scholar 

  • Rogers, C., Shadwick, W.F. (1982) Backlund transformations and their applications. Math. Sci. Eng. pp. 161–300. Academic Press, New York, USA

  • Saad, K.M., Al-Shareef, E.H., Alomari, A.K., Baleanu, D., Aguilar, J.F.G.: On exact solutions for time-fractional Korteweg-de Vries and Korteweg-de Vries-Burgers’ and Burgers’ equations using homotopy analysis transform method. Chinese J. Phys. 63, 149–162 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  • Salam, E.A.B.A., Yousif, E., El-Aasser, M.: Analytical solution of the space-time fractional nonlinear Schrodinger equation. Rep. Math. Phys. 77(1), 19–34 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R.: New exact solutions for the KdV equation with higher order nonlinearity by using the variational method. Comput. Math. Appl. 62(10), 3741–3755 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  • Seadawy, A.R., Ali, K.K., Nuruddeen, R.I.: A variety of soliton solutions for the fractional Wazwaz-Benjamin-Bona-Mahony equations. Res. Phys. 12, 2234–2241 (2019)

    Google Scholar 

  • Tchier, F., Aslan, E.C., Inc, M.: Optical solitons in parabolic law medium: Jacobi elliptic function solution. Nonlinear Dyn. 85(4), 2577–2582 (2016)

    Article  MathSciNet  Google Scholar 

  • Veeresha, P., Prakasha, D.G., Kumar, S.: A fractional model for propagation of classical optical solitons by using nonsingular derivative. Math. Meth. Appl. Sci. 1, 1–8 (2020a)

  • Veeresha, P., Prakasha, D.G., Baskonus, H.M., Yel, G.: An efficient analytical approach for fractional Lalshmanan-Porsezian-Daniel model. Math. Meth. Appl. Sci. 43(7), 4136–4155 (2020b)

    MATH  Google Scholar 

  • Veeresha, P., Baskonus, H.M., Gao, W.: Strong interacting waves in rotating ocean: novel fractional approach. Axioms 10(2), 1–11 (2021)

    Article  Google Scholar 

  • Wazwaz, A.M.: Exact soliton and kink solutions for new (3+1)-dimensional nonlinear modified equations of wave propagation. Open Eng. 7(1), 169–174 (2017)

    Article  Google Scholar 

  • Wazwaz, A.M., Kaur, L.: Optical solitons for nonlinear Schrodinger (NLS) equation in normal dispersive regimes. Optik-Int. J. Light Elect. Opt. 184, 428–435 (2019)

    Article  Google Scholar 

  • Wazwaz, A.M. (2002) Partial differential equations: Method and applications. pp. 1–350. Taylor and Francis

  • Yao, S.W., Zekavatmand, S.M., Rezazadeh, H., Vahidi, J., Chaemi, M.B., Inc, M.: The solitary waves solutions to the Klein-Gordon-Zakharov equations by extended rational methods. AIP Adv. 11(6), 1–10 (2021)

    Article  Google Scholar 

  • Yaslan, H.C., Girgin, A.: New exact solutions for the conformable space-time fractional KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations. J. Taibah Uni. Sci. 13(1), 1–8 (2019)

    Article  Google Scholar 

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Acknowledgements

José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT. G. Fernández-Anaya offer thank for the support given by the DINVP - Universidad Iberoamericana.

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Authors and Affiliations

  1. Department of Mathematics, Hajee Mohammad Danesh Science and Technology University, Dinajpur, Bangladesh

    Md. Tarikul Islam

  2. CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, México

    J. F. Gómez-Aguilar

  3. Department of Applied Mathematics, University of Rajshahi, Rajshahi, Bangladesh

    Md. Ali Akbar

  4. Departamento de Física y Matemáticas, Universidad Iberoamericana, C.P. 01219, Ciudad de México, CDMX, México

    G. Fernández-Anaya

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  1. Md. Tarikul Islam
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  2. J. F. Gómez-Aguilar
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  4. G. Fernández-Anaya
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Contributions

Md. Tarikul Islam: Conceptualization, Methodology, Resources, Formal analysis, Writing-Original draft, Supervision; J.F. Gómez-Aguilar: Conceptualization, Methodology, Writing-review editing, Validation, Final draft preparation, Supervision; Md. Ali Akbar: Conceptualization, Methodology, Software, Writing-Original draft preparation; G. Fernández-Anaya: Conceptualization, Methodology, Writing-review editing, Software, Validation. All authors have read and approved the final manuscript.

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Correspondence to J. F. Gómez-Aguilar or G. Fernández-Anaya.

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Tarikul Islam, M., Gómez-Aguilar, J.F., Ali Akbar, M. et al. Diverse soliton structures for fractional nonlinear Schrodinger equation, KdV equation and WBBM equation adopting a new technique. Opt Quant Electron 53, 669 (2021). https://doi.org/10.1007/s11082-021-03309-9

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