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On the exact soliton solutions and different wave structures to the double dispersive equation

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Abstract

In this manuscript, we are interested to investigate the dynamical behavior of doubly dispersive equation which governs the propagation of nonlinear waves in the elastic Murnaghan’s rod. A variety of solitary wave solutions with unknown parameters is extracted in different shapes such as bright, dark, kink-type, bell-shape, combine and complex soliton, hyperbolic, exponential, and trigonometric function solutions by the employing recently developed methods, namely, new extended direct algebraic method (NEDAM) and generalized Kudryashov method (GKM). Furthermore, we apply the logarithmic transformation, and the ansatz functions method along with symbolic computation. The three waves, double exponential, and homoclinic breather techniques are also utilized to observe the soliton’s interaction phenomenon. A conflict of our results with the considerably-known results are done and it the study states that the solutions reached here are new. In addition, 3D, 2D, and their crossponding contour profiles of earned results are sketched in order to observe their dynamics with the choices of involved parameters. On the bases of achieved results, we may claim that the proposed computational method are direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations. The results obtained for the governing model can be applied to other nonlinear equations. The findings are exceptional and unique in comparison to previous findings in the literature. Furthermore, our findings are first step toward understanding the structure and physical behavior of complicated structures. We anticipate that our results will be highly valuable in better understanding the waves that occur in the solids. We feel that this work is timely and will be of interest to a wide spectrum of experts working on engineering models.

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References

  • Abdelrahman, M.A.E., Ammar1, S.I., Abualnaja, K.M., Inc, M.: New solutions for the unstable nonlinear Schrdinger equation arising in natural science. AIMS Math. 5(3), 1893–1912 (2020)

  • Ahmed, I., Seadawy, A.R., Lu, D.: Kinky breathers, W-shaped and multi-peak solitons interaction in (2 + 1)-dimensional nonlinear Schrodinger equation with Kerr law of nonlinearity. Eur. Phys. J. Plus 134(3), 1–10 (2019)

    Article  Google Scholar 

  • Alquran, M., Sulaiman, T.A., Yusuf, A.: Kink-soliton, singular-kink-soliton and singular-periodic solutions for a new two-mode version of the Burger-Huxley model: applications in nerve fibers and liquid crystals. Opt. Quantam Electron 53(1), 227 (2021)

    Article  Google Scholar 

  • Anjum, N., He, J.H.: Laplace transform: making the variational iteration method easier. Appl. Math. Lett. 92, 134–138 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  ADS  Google Scholar 

  • Bilal, M., Hu, W., Ren, J.: Different wave structures to the Chen-Lee-Liu equation of monomode fibers and its modulation instability analysis. Eur. Phys. J. Plus 136, 385 (2021a)

  • Bilal, M., Ren, J., Younas, U.: Stability analysis and optical soliton solutions to the nonlinear Schrödinger model with efficient computational techniques. Opt. Quantam Electron 53, 406 (2021b)

  • Bilal, M., Younas, U., Ren, J.: Propagation of diverse solitary wave structures to the dynamical soliton model in mathematical physics. Opt. Quantam Electron 53, 522 (2021c)

  • Bilal, M., Younas, U., Ren, J.: Dynamics of exact soliton solutions in the double-chain model of deoxyribonucleic acid. Math. Meth. Appl. Sci. 1–19 (2021d). https://doi.org/10.1002/mma.7631

  • Bilal, M., Younas, U., Ren, J.: Dynamics of exact soliton solutions to the coupled nonlinear system using reliable analytical mathematical approaches. Commun. Theor. Phys. 73, 085005 (2021e)

  • Chen, S.J., Lu, X., Ma, W.X.: Bäcklund transformation, exact solutions and interaction behaviour of the (3+1)-dimensional Hirota-Satsuma-Ito-like equation. Commun. Nonlinear Sci. Numer. Simul. 83, 105135 (2020)

    Article  MathSciNet  Google Scholar 

  • Cattani, C., Rushchitsky, J.: Wavelet and Wave Analysis as Applied to Materials with Micro or Nanostructure. World Scientific (2007) https://doi.org/10.1142/6497

  • Cattani, C., Sulaiman, T.A., Baskonus, H.M., Bulut, H.: Solitons in an inhomogeneous Murnaghan’s rod. Eur. Phys. J. Plus 133, 228 (2018)

  • Dusunceli, F., Celik, E., Askin, M., Bulut, H.: New exact solutions for the doubly dispersive equation using the improved Bernoulli sub-equation function method. Indian J. Phys. 95, 309–314 (2021)

  • Ghanbari, B., Inc, M., Yusuf, A., Baleanu, D.: New solitary wave solutions and stability analysis of the Benney-Luke and the Phi-4 equations in mathematical physics. AIMS Math. 4(6), 1523–1539 (2019)

    Article  MathSciNet  Google Scholar 

  • Hossain, A.K.M.K.S., Akbar, M.A.: Closed form solutions of two nonlinear equation via the enhanced \((\frac{G^{\prime }}{G} )\)-expansion method. Cogent Math. 4 (1), 1355958 (2017)

  • He, C.H., Tang, Y., Ma, W.X., Ma, J.: Interaction phenomena between a lump and other multi-solitons for the (2 + 1)-dimensional BLMP and Ito equations. Nonlinear Dyn. 95, 29–42 (2019)

    Article  Google Scholar 

  • Inc, M.: New compacton and solitary pattern solutions of the nonlinear modied dispersive Klein-Gordon equations. Chaos Solit. Fract. 33, 1275–1284 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  • Inc, M., Esmaates.: Optical soliton solutions for generalized NLSE by using Jacobi elliptic functions. Optoelect. Adv. Meter. Rapid Commun. 9, 1081–1087 (2015)

  • Inc, M., Kilic, B.: Soliton structures of some generalized nonlinear dispersion evolution systems. Proc. Romanian Acad. Ser. A 16, 430–436 (2015)

  • Inc, M., Kilic, B.: The first integral method for the perturbed Wadati-Segur-Ablowitz equation with time dependent coefficient. Kuwait J. Sci. 43, 81–87 (2016)

    MathSciNet  MATH  Google Scholar 

  • Inc, M., Ates, E., Tchier, F.: Optical solitons of the coupled nonlinear Schrdingers equation with spatiotemporal dispersion. Nonlinear Dyn. 85, 1319–1329 (2016a)

  • Inc, M., Kilic, B., Baleanu, D.: Optical soliton solutions of the pulse propagation generalized equation in parabolic-law media with space-modulated coefficients. Optik 127, 1056–1058 (2016b)

  • Kilic, B., Inc, M.: On optical solitons of the resonant Schrdingers equation in optical fibers with dual-power law nonlinearity and time-dependent coefcients. Waves Random Compl. Media 25, 245–251 (2015a)

  • Kilic, B., Inc, M.: The first integral method for the time fractional Kaup-Boussinesq System with time dependent coefficient. Appl. Math. Comput. 254, 70–74 (2015b)

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

  • Kilic, B., Inc, M.: Optical solitons for the Schrodinger-Hirota equation with power law nonlinearity by the Backlund transformation. Optik 138, 64–67 (2017)

    Article  ADS  Google Scholar 

  • Korpinar, Z., Tchier, F., Inc, M.: On optical solitons of the fractional (3+ 1)-dimensional NLSE with conformable derivatives. Front. Phys. 8, 87 (2020)

    Article  Google Scholar 

  • Ma, W.: Interaction solutions to Hirota-Satsuma-Ito equation in (2 + 1)-dimensions. Front. Math. China 14, 619–629 (2019)

    Article  MathSciNet  Google Scholar 

  • Mahak, N., Akram, G.: Exact solitary wave solutions by extended rational sine-cosine and extended rational sinh-cosh techniques. Physica Scripta 94(11), 115212 (2019)

    Article  ADS  Google Scholar 

  • Mahalingam, A., Porsezian, K., Rajan, M.S.M., Uthayakumar, A.: Propagation of dispersion–nonlinearity-managed solitons in an inhomogeneous erbium-doped fiber system. J. Phys. A Math. Theor. 42, 165101 (2009)

  • Malaikah, H.M.: The Adomian decomposition method for solving Volterra-Fredholm integral equation using Maple. Appl. Math. 11, 779–787 (2020)

    Article  Google Scholar 

  • Manafian, J., Ilhan, O., Avazpour, A.L., Alizadeh, A.: N-lump and interaction solutions of localized waves to the(2 + 1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation arise from a model foran incompressible fluid. Math. Methods Appl. Sci. 1–24 (2020). https://doi.org/10.1002/mma.6665

  • Meng, Q.: Rational solutions and interaction solutions for a fourth-order nonlinear generalized Boussinesq water wave equation. Appl. Math. Lett. 110, 106580 (2020)

    Article  MathSciNet  Google Scholar 

  • Osman, M.S., Tariq, K.U., Bekir, A., Elmoasry, A., Elazab, N.S., Younis, M.: Investigation of soliton solutions with different wave structures to the (2+ 1)-dimensional Heisenberg ferromagnetic spin chain equation. Commun. Theor. Phys. 72(3), 035002 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  • Ozdemir, N., Esen, H., Secer, A., Bayram, M., Yusuf, A., Sulaiman, T.A.: Optical soliton solutions to Chen Lee Liu model by the modified extended tanh expansion scheme. Optik 245, 167643 (2021)

    Article  ADS  Google Scholar 

  • Prakash, S.A., Malathi, V., Rajan, M.S.M., Loomba, S.: Controllable pulse width of bright similaritons in a tapered graded index diffraction decreasing wave guide. CHAOS 26, 033115 (2016)

    Article  MathSciNet  ADS  Google Scholar 

  • Rajan, M.S.M.: Dynamics of optical soliton in a tapered erbium-doped fiber under periodic distributed amplification system. Nonlinear Dyn. 85, 599–606 (2016)

    Article  Google Scholar 

  • Rani, M., Ahmed, N., Dragomir, S.S., Mohyud-Din, S.T., Khan, I., Nisar, K.S.: Some newly explored exact solitary wave solutions to nonlinear inhomogeneous Murnaghan’s rod equation of fractional order. J. Taibah Univ. Sci. 15, 97–110 (2021)

  • Rushchitskii, Y.Y., Cattani, C.: Generation of the third harmonics by plane waves in Murnaghan materials. Int. Appl. Mech. 38, 1482 (2002)

    Article  ADS  Google Scholar 

  • Rushchitsky, J.J., Cattani, C.: Similarities and differences between the Murnaghan and Signorini descriptions of the evolution of quadratically nonlinear hyperelastic plane waves. Int. Appl. Mech. 42, 997–1010 (2006)

    Article  ADS  Google Scholar 

  • Rajan, M.S.M., Hakkim, J., Mahalingam, A., Uthayakumar, A.: Dispersion management and cascade compression of femtosecond nonautonomous soliton in birefringent fiber. Eur. Phys. J. D 67, 150 (2013)

    Article  ADS  Google Scholar 

  • Rajan, M.S.M., Mahalingam, A.: Nonautonomous solitons in modified inhomogeneous Hirota equation: soliton control and soliton interaction. Nonlinear Dyn. 79, 2469–2484 (2015)

    Article  MathSciNet  Google Scholar 

  • Rushchitsky, J.J., Saveleva, E.V.: Self-switching of a transverse plane wave propagating through a two-component elastic composite. Int. Appl. Mech. 43, 734–744 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  • Seadawy, A.R., Ali, A., Albarakati, W.A.: Analytical wave solutions of the(2+1)-dimensional first integro-differential Kadomtsev Petviashivili hierarchy equation by using modified mathematical methods. Results Phys. 15, 102775 (2019)

    Article  Google Scholar 

  • Silambarasan, R., Baskonus, H.M., Bulut, H.: Jacobi elliptic function solutions of the double dispersive equation in the Murnaghan’s rod. Eur. Phys. J. Plus 134, 125 (2019)

  • Seadawy, A.R., Lu, D., Nasreen, N.: Construction of solitary wave solutions of some nonlinear dynamical system arising in nonlinear water wave models. Indian J. Phys. 94, 1785–1794 (2020)

    Article  ADS  Google Scholar 

  • Seadawy, A.R., Rashidy, K.E.: Application of the extension exponential rational function method for higher-dimensional Broer-Kaup-Kupershmidt dynamical system. Modern Phys. Lett. A 35(01), 1950345 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  • Shi, D., Zhang, Y.: Diversity of exact solutions to the conformable space-time fractional MEW equation. Appl. Math. Lett. 99, 105994 (2020)

    Article  MathSciNet  Google Scholar 

  • Tian, S.F.: Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation. Appl. Math. Lett. 100, 106056 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  • Yildirim, Y.: Optical solitons of Biswas-Arshed equation by trial equation technique. Optik 182, 876–883 (2019)

    Article  ADS  Google Scholar 

  • Yue, C., Khater, M.M.A., Inc, M., Attia, R.A.M., Lu, D.: Abundant analytical solutions of the fractional nonlinear (2+ 1)-dimensional BLMP equation arising in incompressible fluid. Int. J. Modern Phys. B 34(09), 2050084 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  • Younas, U., Ren, J.: Investigation of exact soliton solutions in magneto-optic waveguides and its stability analysis. Results Phys. 21, 103816 (2021)

    Article  Google Scholar 

  • Younas, U., Bilal, M., Ren, J.: Propagation of the pure-cubic optical solitons and stability analysis in the absence of chromatic dispersion. Opt. Quantam Electron. 53, 490 (2021)

    Article  Google Scholar 

  • Yusuf, A., Sulaiman, T.A., Inc, M., Bayram, M.: Breather wave, lump-periodic solutions and some other interaction phenomena to the Caudrey-Dodd-Gibbon equation. Eur. Phys. J. Plus 135(7), 1–8 (2020)

    Article  Google Scholar 

  • Younis, M., Younas, U., Rehman, S.U., Bilal, M., Waheed, A.: Optical bright-dark and Gaussian soliton with third order dispersion. Optik 134, 233–238 (2017)

    Article  ADS  Google Scholar 

  • Younis, M., Rizvi, S.T.R., Baleanu, D., Iqbal, H.: Lump and rogue wave solutions for the Broer-Kaup-Kupershmidt system. Chinese J. Phys. 68, 19–27 (2020)

    Article  MathSciNet  ADS  Google Scholar 

  • Zayed, E.M.E., Al-Nowehy, A.G., Elshater, M.E.M.: New \(\Phi ^{6}\)-model expansion method and its applications to the resonant nonlinear Schrodinger equation with parabolic law nonlinearity. Eur. Phys. J. Plus 133, 417 (2018)

    Article  Google Scholar 

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Acknowledgements

The authors would like to acknowledge the financial support provided for this research via the National Natural Science Foundation of China (52071298), and Zhong Yuan Science and Technology Innovation Leadership Program (214200510010). They also thank the reviewers for their valuable reviews and kind suggestions.

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Authors and Affiliations

  1. Henan Academy of Big Data/School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, China

    Usman Younas, Muhammad Bilal & Jingli Ren

  2. Department of Computer Engineering, Biruni University Istanbul, Zeytinburnu, Turkey

    Tukur Abdulkadir Sulaiman & Abdullahi Yusuf

  3. Faculty of Science, Federal University Dutse, Dutse, Jigawa, Nigeria

    Tukur Abdulkadir Sulaiman & Abdullahi Yusuf

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  1. Usman Younas
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  2. Muhammad Bilal
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  3. Tukur Abdulkadir Sulaiman
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  5. Abdullahi Yusuf
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Correspondence to Jingli Ren.

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Younas, U., Bilal, M., Sulaiman, T.A. et al. On the exact soliton solutions and different wave structures to the double dispersive equation. Opt Quant Electron 54, 71 (2022). https://doi.org/10.1007/s11082-021-03445-2

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