In this paper, the improved $\tan\left(\Phi(\xi)/2\right)$-expansion method (ITEM) is proposed to obtain the fractional Biswas-Milovic equation. The exact particular solutions contain four types: hyperbolic function solution, trigonometric function solution, exponential solution, and rational solution. We obtained further solutions compared with other methods, such as [2]. Recently, this method has been developed for searching exact travelling wave solutions of nonlinear partial differential equations. These solutions might play an important role in nonlinear optics and physics. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving problems in nonlinear optics.
[1] M. F. Aghdaei and J. Manafian, Optical soliton wave solutions to the resonant Davey-Stewartson system, Opt. Quant. Electron, 48 (2016), 1-33.
[2] S. Ahmadiana and M. T. Darvishi, A new fractional Biswas-Milovic model with its periodic soliton solutions, Optik-International Journal for Light and Electron Optics, 38 (2016), 3763-3767.
[3] I. Ahmed, C. Mu, and F. Zhang, Exact solution of the Biswas-Milovic equation by Adomian decomposition method, Int. J. Appl. Math. Research, 2 (2011), 418-422.
[4] N. H. Ali, S. A. Mohammed, and J. Manafian, Study on the simplified MCH equation and the combined KdVmKdV equations with solitary wave solutions, Partial Diff. Eq. Appl. Math., 9 (2024), 100599.
[5] A. Biswas, C. Cleary, J. E. Watson, and D. Milovic, Optical soliton perturbation with time-dependent coefficients in a log law media, Appl. Math. Comput., 217 (2010), 2891-2894.
[6] A. Biswas and D. Milovic, Bright and dark solitons of the generalized nonlinear Schrodinger’s equation, Commu. Nonlinear Sci. Num. Simul., 15 (2010), 1473-1484.
[7] A. Biswas and D. Milovic, Optical solitons with log law nonlinearity, Commu. Nonlinear Sci. Num. Simul., 15 (2010), 3763-3767.
[8] A. Biswas and S. Konar, Introduction to Non-Kerr Law Optical Solitons, CRC Press, Boca Raton, FL, USA, (2006).
[9] M. Caputo, Elasticita e dissipazione, Zani-Chelli, Bologna, 1969.
[10] H. Chen, A. Shahi, G. Singh, J. Manafian, B. Baharak Eslami, and N. A. Alkader, Behavior of analytical schemes with non-paraxial pulse propagation to the cubic-quintic nonlinear Helmholtz equation, Math. Comput. Simul., 220 (2024), 341-356.
[11] L. Debnath, Fractional integrals and fractional differential equations in fluid mechanics, Frac. Calc. Appl. Anal., 6 (2003), 119-155.
[12] M. Dehghan and J. Manafian, The solution of the variable coefficients fourth–order parabolic partial differential equations by homotopy perturbation method, Z. Naturforsch, 64a (2009), 420-430.
[13] M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Num. Meth. Partial Diff.l Eq. J., 26 (2010), 448-479.
[14] M. Dehghan, J. Manafian, and A. Saadatmandi, Application of the Exp-function method for solving a partial differential equation arising in biology and population genetics, Int J. Num. Methods Heat Fluid Flow, 21 (2011), 736-753.
[15] M. Dehghan, J. Manafian, and A. Saadatmandi, Application of semi–analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses, Math. Meth. Appl. Sci., 33 (2010), 1384-1398.
[16] Y. Gu, S. Malmir, J. Manafian, O. A. Ilhan, A. A. Alizadeh, and A. J. Othman, Variety interaction between k-lump and k-kink solutions for the (3+1)-D Burger system by bilinear analysis, Results Phys., 43 (2022), 106032.
[17] H. Jafari, A. Soorakia, and C. M. Khalique, Dark solitons of the Biswas-Milovic equation by the first integral method, Optik, 124 (2013), 3929-3932.
[18] C. M. Khalique, Stationary solutions for the Biswas-ilovic equation, Appl. Math. Comput., 217 (2011) 7400-7404.
[19] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Amsterdam, 2006.
[20] X. G. Luo, Q. B. Wub, and B. Q. Zhang, Revisit on partial solutions in the Adomian decomposition method: Solving heat and wave equations, J. Math. Anal. Appl., 321 (2006), 353-363.
[21] R. Kohla, R. Tinaztepeb, and A. Chowdhury, Soliton perturbation theory of Biswas-Milovic equation, Optik, 125 (2014), 1926-1936.
[22] M. Lakestani, J. Manafian, A. R. Najafizadeh, and M. Partohaghighi, Some new soliton solutions for the nonlinear the fifth-order integrable equations, Comput. Meth. Diff. Equ., 10(2) (2022).
[23] J. Manafian, On the complex structures of the Biswas-Milovic equation for power, parabolic and dual parabolic law nonlinearities, Eur. Phys. J. Plus, 130 (2015), 1-20.
[24] J. Manafian, Optical soliton solutions for Schrodinger type nonlinear evolution equations by the tan(ϕ/2)-expansion method, Optik-Int. J. Elec. Opt., 127 (2016), 4222-4245.
[25] J. Manafian, M. F. Aghdaei, and M. Zadahmad, Analytic study of sixth-order thin-film equation by tan(ϕ/2)expansion method, Opt. Quant. Electron, 48 (2016), 1-16.
[26] J. Manafian, L. A. Dawood, and M. Lakestani, New solutions to a generalized fifth-order KdV like equation with prime number p = 3 via a generalized bilinear differential operator, Partial Diff. Eq. Appl. Math., 9 (2024), 100600.
[27] J. Manafian and M. Lakestani, Application of tan(ϕ/2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearityJalil, Optik, 127 (2016), 2040-2054.
[28] J. Manafian and M. Lakestani, A new analytical approach to solve some the fractional-order partial differential equations, Indian J. Phys., 90 (2016), 1-16.
[29] J. Manafian and M. Lakestani, Optical solitons with Biswas-Milovic equation for Kerr law nonlinearity, Eur. Phys. J. Plus, 130 (2015), 1-12.
[30] J. Manafian and M. Lakestani, Application of tan(ϕ/2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity, Optik-Int. J. Elec. Opt., 127 (2016), 2040-2054.
[31] J. Manafian and M. Lakestani, Dispersive dark optical soliton with Tzitz´eica type nonlinear evolution equations arising in nonlinear optics, Opt. Quant. Electron, 48 (2016), 1-32.
[32] J. Manafian and M. Lakestani, Solitary wave and periodic wave solutions for Burgers, Fisher, Huxley and combined forms of these equations by the G′/G-expansion method, Pramana- J. Phys., 130 (2015), 31-52.
[33] J. Manafian, M. Lakestani, and A. Bekir, Comparison between the generalized tanh-coth and the G′/G-expansion methods for solving NPDE’s and NODE’s, Pramana. J. Phys., 87 (2016), 1-14.
[34] J. Manafian, M. Lakestani, and A. Bekir, Study of the analytical treatment of the (2+1)-dimensional Zoomeron, the Duffing, and the SRLW equations via a new analytical approach, Int. J. Appl. Comput. Math., 2 (2016), 243-268.
[35] J. Manafian and M. Lakestani, New improvement of the expansion methods for solving the generalized Fitzhugh-Nagumo equation with time-dependent coefficients, Int. J. Eng. Math., 2015 (2015), 1-35.
[36] J. Manafian and M. Lakestani, N-lump and interaction solutions of localized waves to the (2+ 1)- dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation, J. Geom. Phys., 150 (2020), 103598.
[37] J. Manafian and M. Lakestani, Application of tan(ϕ/2)-expansion method for solving the Biswas-Milovic equation for Kerr law nonlinearity, Optik, 127(4) (2016), 2040-2054.
[38] J. Manafian and M. Lakestani, Optical soliton solutions for the Gerdjikov-Ivanov model via tan(ϕ/2)-expansion method, Optik, 127(20) (2016), 9603-9620.
[39] J. Manafian and M. Lakestani, Abundant soliton solutions for the Kundu-Eckhaus equation via tan(ϕ(ξ))expansion method, Optik, 127(14) (2016), 5543-5551.
[40] M. Mirzazadeh and M. Eslami, Exact multisoliton solutions of nonlinear Klein-Gordon equation in 1 + 2 dimensions, The Eur. Phys. J. Plus, 128 (2015), 1-9.
[41] S. R. Moosavi, N. Taghizadeh, and J. Manafian, Analytical approximations of one-dimensional hyperbolic equation with non-local integral conditions by reduced differential transform method, Comput. Meth. Diff. Equ., 8(3) (2020), 537-552.
[42] K. Oldham, Fractional differential equations in electrochemistry, Adv. Eng. Softw., 41 (2010), 9-17.
[43] B. Sturdevant, Topological 1-soliton solution of the Biswas-Milovic equation with power law nonlinearity, Nonlinear Analysis; Real World Appl., 11 (2010), 2871-2874.
[44] G. O. Young, Definition of physical consistent damping laws with fractional derivatives, Z. Angew. Math. Mech., 75 (1995), 623-635.
[45] Z. Y. Zhang, Z. H. Liu, X. J. Miao, and Y. Z. Chen, Qualitative analysis and traveling wave solutions for the perturbed nonlinear Schrodinger’s equation with Kerr law nonlinearity, Phys. Let. A, 375 (2011), 1275-1280.
[46] S. Zhang, J. Manafian, O. A. Ilhan, A. T. Jalil, Y. Yasin, and M. A. Gatea, Nonparaxial pulse propagation to the cubic-quintic nonlinear Helmholtz equation, International Journal of Modern Physics B, 38(8) (2024), 2450117.
[47] M. Zhang, X. Xie, J. Manafian, O. A. Ilhan, and G. Singh, Characteristics of the new multiple rogue wave solutions to the fractional generalized CBS-BK equation, J. Adv. Res., 38 (2022), 131-142.
[48] N. Zhao, J. Manafian, O. A. Ilhan, G. Singh, and R. Zulfugarova, Abundant interaction between lump and k-kink, periodic and other analytical solutions for the (3+1)-D Burger system by bilinear analysis, Int. J. Modern Phys. B, 35(13) (2021), 2150173.
[49] X. Zhao, L. Wang, and W. Sun, The repeated homogeneous balance method and its applications to nonlinear partial differential equations, Chaos Solitons Fract., 28 (2006), 448-453.
Fugarov, D. , Dengaev, A. , Drozdov, I. , Shishulin, V. and Ostrovskaya, A. (2025). Application of $tan(\phi/2)$-expansion method for solving the fractional Biswas-Milovic equation for Kerr law nonlinearity. Computational Methods for Differential Equations, 13(4), 1408-1424. doi: 10.22034/cmde.2024.61349.2636
MLA
Fugarov, D. , , Dengaev, A. , , Drozdov, I. , , Shishulin, V. , and Ostrovskaya, A. . "Application of $tan(\phi/2)$-expansion method for solving the fractional Biswas-Milovic equation for Kerr law nonlinearity", Computational Methods for Differential Equations, 13, 4, 2025, 1408-1424. doi: 10.22034/cmde.2024.61349.2636
HARVARD
Fugarov, D., Dengaev, A., Drozdov, I., Shishulin, V., Ostrovskaya, A. (2025). 'Application of $tan(\phi/2)$-expansion method for solving the fractional Biswas-Milovic equation for Kerr law nonlinearity', Computational Methods for Differential Equations, 13(4), pp. 1408-1424. doi: 10.22034/cmde.2024.61349.2636
CHICAGO
D. Fugarov , A. Dengaev , I. Drozdov , V. Shishulin and A. Ostrovskaya, "Application of $tan(\phi/2)$-expansion method for solving the fractional Biswas-Milovic equation for Kerr law nonlinearity," Computational Methods for Differential Equations, 13 4 (2025): 1408-1424, doi: 10.22034/cmde.2024.61349.2636
VANCOUVER
Fugarov, D., Dengaev, A., Drozdov, I., Shishulin, V., Ostrovskaya, A. Application of $tan(\phi/2)$-expansion method for solving the fractional Biswas-Milovic equation for Kerr law nonlinearity. Computational Methods for Differential Equations, 2025; 13(4): 1408-1424. doi: 10.22034/cmde.2024.61349.2636
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