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On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics

  • Received: 05 July 2020 Accepted: 17 August 2020 Published: 24 August 2020
  • MSC : 35A09

  • Nonlinear equations personate a consequential role in scientific fields such as nonlinear optics, solid state physics and quantum field theory. This article studies the Tzitzéica-Dodd-Bullough-Mikhailov and Tzitzéica-type equations that appear in nonlinear optics. The Painlevé and traveling wave transformations both play a key role in revamping the aforementioned equations into nonlinear ordinary differential equations. Then, the simple ansatz approach is followed to seize complex singular, complex bright solitons and other kink type solutions. The existing literature unveils that this study is a novel contribution in the literature and the proposed approach is forthright and simple to implement for solving nonlinear problems. This approach has no extra condition as compared to many other techniques have. We also verify and interpret graphically the secured solutions through symbolic software Mathematica and MatLab respectively.

    Citation: Yongsheng Rao, Asim Zafar, Alper Korkmaz, Asfand Fahad, Muhammad Imran Qureshi. On Tzitéeica type nonlinear equations for multiple soliton solutions in nonlinear optics[J]. AIMS Mathematics, 2020, 5(6): 6580-6593. doi: 10.3934/math.2020423

    Related Papers:

  • Nonlinear equations personate a consequential role in scientific fields such as nonlinear optics, solid state physics and quantum field theory. This article studies the Tzitzéica-Dodd-Bullough-Mikhailov and Tzitzéica-type equations that appear in nonlinear optics. The Painlevé and traveling wave transformations both play a key role in revamping the aforementioned equations into nonlinear ordinary differential equations. Then, the simple ansatz approach is followed to seize complex singular, complex bright solitons and other kink type solutions. The existing literature unveils that this study is a novel contribution in the literature and the proposed approach is forthright and simple to implement for solving nonlinear problems. This approach has no extra condition as compared to many other techniques have. We also verify and interpret graphically the secured solutions through symbolic software Mathematica and MatLab respectively.


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    [1] M. Tzitzeica, Sur une nouvelle classe des surfaces, CR Acad. Sci. Paris, 150 (1910), 955-956.
    [2] W. Rui, Exact traveling wave solutions for a nonlinear evolution equation of generalized tzitzéica-dodd-bullough-mikhailov type, J. Appl. Math., 2013 (2013), 1-14.
    [3] A. M. Wazwaz, The tanh method: Solitons and periodic solutions for the dodd-bullough-mikhailov and the tzitzeica-dodd-bullough equations, Chaos Soliton Fract., 25 (2005), 55-63. doi: 10.1016/j.chaos.2004.09.122
    [4] R. K. Dodd, R. K. Bollough, Polynomial conserved densities for the sine-gordon equations, Proc. R. Soc. Lond. A, 352 (1977), 481-503. doi: 10.1098/rspa.1977.0012
    [5] R. Conte, M. Musette, A. M. Grunland, Backlund transformation of partial differential equations from the painleve-gambier classification ii. tzitzeica equation, J. Math. Phys., 40 (1999), 2092-2106. doi: 10.1063/1.532853
    [6] J. Y. Zhu, X. G. Geng, Darboux transformation for tzitzeica equation, Commun. Theor. Phys., 45 (2006), 577-580. doi: 10.1088/0253-6102/45/4/001
    [7] A. V. Mikhailov, The reduction problem and the inverse scattering method, Physica, 1 (1981), 73-117.
    [8] O. H. El-Kalaawy, Exact soliton solutions for some nonlinear partial differential equations, Chaos Soliton Fract., 14 (2002), 547-552. doi: 10.1016/S0960-0779(01)00217-X
    [9] K. Khan, M. A. Akbar, Exact and solitary wave solutions for the tzitzeica-dodd-bullough and the modified kdv-zakharov-kuznetsov equations using the modified simple equation method, Ain Shams Eng. J., 4 (2013), 903-909. doi: 10.1016/j.asej.2013.01.010
    [10] R. Abazari, The (g'/g)-expansion method for tzitzéica type nonlinear evolution equations, Math. Comput. Model., 52 (2010), 1834-1845. doi: 10.1016/j.mcm.2010.07.013
    [11] J. Manafian, M. Lakestani, Dispersive dark optical soliton with tzitzéica type nonlinear evolution equations arising in nonlinear optics, Opt. Quant. Electron., 48 (2016), 116.
    [12] A. Korkmaz, K. Hosseini, Exact solutions of a nonlinear conformable time fractional parabolic equation with exponential nonlinearity using reliable methods, Opt. Quant. Electron., 49 (2017), 1-10. doi: 10.1007/s11082-016-0848-8
    [13] K. Hosseini, Z. Ayati, R. Ansari, New exact traveling wave solutions of the tzitzéica type equations using a novel exponential rational function method, Optik, 148 (2017), 85-89. doi: 10.1016/j.ijleo.2017.08.030
    [14] K. Hosseini, Z. Ayati, R. Ansari, New exact solution of the tzitzéica type equations in nonlinear optics using the expa function method, J. Mod. Opt., 65 (2018), 847-851. doi: 10.1080/09500340.2017.1407002
    [15] D. Kumar, K. Hosseini, F. Samadani, The sine-gordon expansion method to look for the traveling wave solutions of the tzitzéica type equations in nonlinear optics, Optik, 149 (2017), 439-446. doi: 10.1016/j.ijleo.2017.09.066
    [16] S. M. M. Alizamini, H. Rezazadeh, M. Eslami, et al. New extended direct algebraic method for the Tzitzica type evolution equations arising in nonlinear optics, Comp. Meth. Diff. Eq., 8 (2020): 28-53.
    [17] A. H. Arnous, A. R. Seadawy, R. T. Alqahtani, et al. Optical solitons with complex ginzburg-landau equation by modified simple equation method, Optik, 144 (2017), 475-480. doi: 10.1016/j.ijleo.2017.07.013
    [18] H. Rezazadeh, A. Korkmaz, M. Eslami, et al. Traveling wave solution of conformable fractional generalized reaction duffing model by generalized projective riccati equation method, Opt. Quant. Electron., 50 (2018), 1-13. doi: 10.1007/s11082-017-1266-2
    [19] K. Hosseini, P. Mayeli, R. Ansari, Bright and singular soliton solutions of the conformable time-fractional klein-gordon equations with different nonlinearities, Wave Random Complex., 28 (2018), 426-434.
    [20] A. Zafar, M. Raheel, A. Bekir, Exploring the dark and singular soliton solutions of biswas-arshed model with full nonlinear form, Optik, 204 (2020), 164133.
    [21] A. Korkmaz, Complex wave solutions to mathematical biology models i: Newell-whitehead-segel and zeldovich equations, J. Comput. Nonlin. Dyn., 13 (2018), 1-7.
    [22] A. Korkmaz, Exact solutions to (3+1) conformable time fractional jimbo-miwa, zakharov-kuznetsov and modified zakharov-kuznetsov equations, Commun. Theor. Phys., 67 (2017), 479.
    [23] Z. Ayati, K. Hosseini, M. Mirzazadeh, Application of kudryashov and functional variable methods to the strain wave equation in microstructured solids, Nonlinear Engineering, 6 (2017), 25-29.
    [24] K. Hosseini, J. Manafian, F. Samadani, et al. Resonant optical solitons with perturbation terms and fractional temporal evolution using improved tan(φ(η)/2)-expansion method and exp function approach, Optik, 158 (2018), 933-939. doi: 10.1016/j.ijleo.2017.12.139
    [25] K. Hosseini, P. Mayeli, D. Kumar, New exact solutions of the coupled sine- gordon equations in nonlinear optics using the modified kudryashov method, J. Mod. Optic., 65 (2018), 361-364. doi: 10.1080/09500340.2017.1380857
    [26] K. Hosseini, F. Samadani, D. Kumar, et al. New optical solitons of cubic-quartic nonlinear schr'odinger equation, Optik, 157 (2018), 1101-1105. doi: 10.1016/j.ijleo.2017.11.124
    [27] J. Vahidi, A. Zafar, A. Bekir, et al. The functional variable method to find new exact solutions of the nonlinear evolution equations with dual-power-law nonlinearity, Int. J. Nonlin. Sci. Num., (2020), article no. 10.1515/ijnsns-2019-0064.
    [28] M. S. Osman, H. Rezazadeh, M. Eslami, Traveling wave solutions for (3+1) dimensional conformable fractional zakharov-kuznetsov equation with power law nonlinearity, Nonlinear Engineering, 8 (2019), 559-567. doi: 10.1515/nleng-2018-0163
    [29] M. S. Osman., New analytical study of water waves described by coupled fractional variant boussinesq equation in fluid dynamics, Pramana, 93 (2019), 26.
    [30] V. S. Kumar, H. Rezazadeh, M. Eslami, et al. Jacobi Elliptic Function Expansion Method for Solving KdV Equation with Conformable Derivative and Dual-Power Law Nonlinearity, Int. J. Appl. Comput. Math., 5 (2019), 127.
    [31] A. Zafar, The expa function method and the conformable time-fractional kdv equations, Nonlinear Engineering, 8 (2019), 728-732. doi: 10.1515/nleng-2018-0094
    [32] H. Rezazadeh, A. Bekir, A. Zafar, et al. Exact solutions of (3 + 1)-dimensional fractional mkdv equations in conformable form via exp(-φ(τ))-expansion method, SN App. Sciences, 11 (2019), article no. 1436.
    [33] K. Hosseini, A. Zabihi, F. Samadani, et al. New explicit exact solutions of the unstable nonlinear schrödinger's equation using the expa and hyperbolic function methods, Opt. Quant. Electron., 50 (2018), article no. 82.
    [34] A. Zafar, A. Seadawy, The conformable space-time fractional mkdv equations and their exact solutions, J. King Saud Uni. Sci., 31 (2019), 1478-1484. doi: 10.1016/j.jksus.2019.09.003
    [35] A. H. Soliman, Abd-Allah Hyder, Closed-form solutions of stochastic KdV equation with generalized conformable derivatives, Phys. Scripta, 95 (2020), 065219.
    [36] Abd-Allah Hyder, A. H. Soliman, Exact solutions of space-time local fractal nonlinear evolution equations: A generalized conformable derivative approach, Results Phy., 17 (2020), 103135.
    [37] Abd-Allah Hyder, M. A. Barakat, General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics, Phys. Scripta, 95 (2020), 045212.
    [38] H. A. Ghany, Abd-Allah Hyder, M. Zakarya, Exact solutions of stochastic fractional Korteweg de-Vries equation with conformable derivatives, Chinese Phys. B, 29 (2020), 030203.
    [39] A. Hyder, White noise theory and general improved Kudryashov method for stochastic nonlinear evolution equations with conformable derivatives, Adv. Differ. Equ., 236 (2020).
    [40] D. Lu, A. Seadawy, M. Arshad, Elliptic function solutions and travelling wave solutions of nonlinear dodd-bullough-mikhailov, two-dimensional sine-gordon and coupled schrödinger-kdv dynamical models, Results Phys., 10 (2018), 995-1005. doi: 10.1016/j.rinp.2018.08.001
    [41] A. Seadawy, S. Z. Alamri, Mathematical methods via the nonlinear two-dimensional water waves of olver dynamical equation and its exact solitary wave solutions, Results Phys., 8 (2018), 286-291. doi: 10.1016/j.rinp.2017.12.008
    [42] D. Lu, A. Seadawy, M. M. A. Khater, Bifurcations of new multi soliton solutions of the van der waals normal form for fluidized granular matter via six different methods, Results Phys., 7 (2017), 2028-2035. doi: 10.1016/j.rinp.2017.06.014
    [43] M. Iqbal, A. Seadawy, O. Khalil, et al. Propagation of long internal waves in density stratified ocean for the (2+1)-dimensional nonlinear nizhnik-novikov-vesselov dynamical equation, Results Phys., 16 (2020), article no. 102838.
    [44] A. Javid, N. Raza, M. S. Osman, Multi-solitons of thermophoretic motion equation depicting the wrinkle propagation in substrate-supported graphene sheets, Commun. Theor. Phys., 71 (2019), 362.
    [45] M. S. Osman, A. M. Wazwaz, A general bilinear form to generate different wave structures of solitons for a (3+1)-dimensional boiti-leon-manna-pempinelli equation, Math. Method Appl. Sci., 42, 2019.
    [46] M. S. Osman, D. Lu, M. M. A. Khater., A study of optical wave propagation in the nonautonomous schrödinger-hirota equation with power-law nonlinearity, Results Phys., 13 (2019), 102157.
    [47] D. Lu, M. S. Osman, M. M. A. Khater, et al. Analytical and numerical simulations for the kinetics of phase separation in iron (fe-cr-x (x=mo,cu)) based on ternary alloys, Phys. A, 537 (2020), 122634.
    [48] K. K. Ali, C. Cattani, J. F. Gómez-Aguilar, et al. Analytical and numerical study of the DNA dynamics arising in oscillator-chain of Peyrard-Bishop model, Chaos Soliton Fract., 139 (2020), 110089.
    [49] Y. Ding, M. S. Osman, A. M. Wazwaz, Abundant complex wave solutions for the nonautonomous Fokas-Lenells equation in presence of perturbation terms, Optik, 181 (2019), 503-513. doi: 10.1016/j.ijleo.2018.12.064
    [50] D. Lu, K. U. Tariq, M. S. Osman, et al. New analytical wave structures for the (3 + 1)-dimensional Kadomtsev-Petviashvili and the generalized Boussinesq models and their applications, Results Phys., 14 (2019), 102491.
    [51] I. Hamdy, A. Gawad, M. Osman, On shallow water waves in a medium with time-dependent dispersion and nonlinearity coefficients, J. Adv. Res., 6 (2015), 593-599. doi: 10.1016/j.jare.2014.02.004
    [52] M. S. Osman, Analytical study of solitons to benjamin-bona-mahony-peregrine equation with power law nonlinearity by using three methods, U.P.B. Sci. Bull., Series A, 80 (2018).
    [53] R. Nuruddeen, Multiple soliton solutions for the (3+1) conformable space-time fractional modified korteweg-de-vries equations, J. Ocean Eng. Sci., 3 (2018), 11-18. doi: 10.1016/j.joes.2017.11.004
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