A study on the solutions of (3+1) conformal time derivative generalized q-deformed Sinh-Gordon equation

This article is about examining the solutions of the (3+1) conformal time derivative generalized q-deformed Sinh-Gordon equation. The integration method used to reach the solutions of the equation is the generalized exponential rational function method. In this article, the process of examining the solutions goes step by step, first the basic steps of the proposed method are given, then the reduction of the equation is examined, and then the solutions are obtained by applying the method. To perceive the physical phenomena, 2D and 3D graphical patterns of some of solutions obtained in this study are plotted by using computer programming. The worked-out solutions ascertained that the suggested method is effectual, simple and direct.

Keywords:

Conformal time derivative, The generalized exponential rational function method The generalized q- deformed Sinh–Gordon equation,

PDF

___

[1]. Kurt, A. 2020. New analytical and numerical results for fractional Bogoyavlensky-Konopelchenko equation arising in fluid dynamics. Applied Mathematics-A Journal of Chinese Universities; 35, 101-112.
[2]. Wei, CC, Tian B, Yang, DY, Liu, SH. 2023. Jacobian-elliptic-function and rogue-periodic-wave solutions of a high-order nonlinear Schrödinger equation in an inhomogeneous optical fiber. Chin. J. Phys.; 81, 354–361.
[3]. Poonia, M, Singh, K. 2023. Exact solutions of nonlinear dynamics of microtubules equation using the methods of first integral and (G’/G) expansion. Asian-Eur. J. Math.; 16(01), 2350007.
[4]. Younas, U, Baber, MZ, Yasin, MW, Sulaiman, TA, Ren, J. 2022. The generalized higher-order nonlinear Schrödinger equation: optical solitons and other solutions in fiber optics. Int. J. Mod. Phys. B; 2350174.
[5]. Tarla, S, Ali, KK, Yilmazer, R, Osman, MS. 2022. On dynamical behavior for optical solitons sustained by the perturbed Chen–Lee–Liu model. Communications in Theoretical Physics; 74(7), 075005.
[6]. Iqbal, MS, Seadawy, AR, Baber, MZ, Qasim, M. 2022. Application of modified exponential rational function method to Jaulent–Miodek system leading to exact classical solutions. Chaos, Solitons & Fractals; 164, 112600.
[7]. Taghizadeh, N, Akbari, M, Esmaeelnejhad, P. 2017. Application of Bernoulli sub-ODE method for finding travelling wave solutions of Schrodinger equation power law nonlinearity. Applications and Applied Mathematics: An International Journal (AAM); 12(1), 38.
[8]. Li, L, Yu, F, Duan, C. 2020. A generalized nonlocal Gross-Pitaevskii (NGP) equation with an arbitrary time-dependent linear potential. Applied Mathematics Letters; 110, 106584.
[9]. Younas, U, Sulaiman, TA, Ren, J. 2022. Dynamics of optical pulses in fiber optics with stimulated Raman scattering effect. International Journal of Modern Physics B; 2350080.
[10]. Salas, AH, Gomez S, CA. 2010. Application of the Cole-Hopf transformation for finding exact solutions to several forms of the seventh-order KdV equation. Mathematical Problems in Engineering; 2010.
[11]. San, S, Seadawy, AR, Yasar, E. 2022. Optical soliton solution analysis for the (2+ 1) dimensional Kundu-Mukherjee-Naskar model with local fractional derivatives. Optical and Quantum Electronics; 54(7), 442.
[12]. Hosseini, K, Mirzazadeh, M, Akinyemi, L, Baleanu, D, Salahshour, S. 2022. Optical solitons to the Ginzburg-Landau equation including the parabolic nonlinearity. Optical and Quantum Electronics; 54(10), 631.
[13]. Hosseini, K, Mirzazadeh, M, Baleanu, D, Raza, N, Park, C, Ahmadian, A, Salahshour, S. 2021. The generalized complex Ginzburgâ “Landau model and its dark and bright soliton solutions. The European Physical Journal Plus; 136(7), 1-12.
[14]. Wang, KJ, Si, J. (2023). Diverse optical solitons to the complex Ginzburgâ “Landau equation with Kerr law nonlinearity in the nonlinear optical fiber. The European Physical Journal Plus; 138(3), 187.
[15]. Bullough, RK, Pilling, DJ, Timonen, J. 1986. Quantum and classical statistical mechanics of the sinh-Gordon equation. Journal of Physics A: Mathematical and General; 19(16), L955.
[16]. Wazwaz, AM. 2012. One and two soliton solutions for the sinhâ “Gordon equation in (1+ 1),(2+ 1) and (3+ 1) dimensions. Applied Mathematics Letters; 25(12), 2354-2358.
[17]. Wazwaz, AM. 2005. The tanh method: exact solutions of the sine-Gordon and the sinh-Gordon equations. Applied Mathematics and Computation; 167(2): 1196-1210.
[18]. Malomed, BA, Mihalache, D. 2019. Nonlinear waves in optical and matter-wave media: a topical survey of recent theoretical and experimental results. Rom. J. Phys; 64, 106.
[19]. Eleuch, H. 2018. Some analytical solitary wave solutions for the generalized q-deformed Sinh-Gordon equation. Advances in Mathematical Physics; 2018.
[20]. Victor, K, Pokman, C. 2002. Quantum calculus.
[21]. Arai, A. 1991. Exactly solvable supersymmetric quantum mechanics. Journal of Mathematical Analysis and Applications; 158(1), 63-79.
[22]. Kibble, TWB. 2006. Topological Defects and Their Homotopy Classification.
[23]. Alrebdi, HI, Raza, N, Arshed, S, Butt, AR, Abdel-Aty, AH, Cesarano, C, Eleuch, H. 2022. A variety of new explicit analytical soliton solutions of q-deformed sinh-gordon in (2+ 1) dimensions. Symmetry; 14(11), 2425.
[24]. Khalil, R, Al Horani, M, Yousef, A, Sababheh, M. 2014. A new definition of fractional derivative. Journal of computational and applied mathematics; 264, 65-70.
[25]. Ali, KK, Al-Harbi, N, Abdel-Aty, AH. 2023. Traveling wave solutions to (3+ 1) conformal time derivative generalized q-deformed Sinh-Gordon equation. Alexandria Engineering Journal; 65, 233-243.
[26]. Ghanbari, B, Inc, M. 2018. A new generalized exponential rational function method to find exact special solutions for the resonance nonlinear Schrodinger equation. The European Physical Journal Plus; 133(4): 142.
[27]. Tarla, S, Ali, KK, Yilmazer, R, Osman, MS. 2022. Propagation of solitons for the Hamiltonian amplitude equation via an analytical technique. Modern Physics Letters B; 36(23): 2250120.
[28]. Ali, KK, Yilmazer, R, Bulut, H, Akturk, T, Osman, MS. 2021. Abundant exact solutions to the strain wave equation in micro-structured solids. Modern Physics Letters B; 35(26): 2150439.
[29]. Ghanbari, B, Liu, JG. 2020. Exact solitary wave solutions to the (2+ 1)-dimensional generalised Camassa-Holm-Kadomtsev-Petviashvili equation. Pramana; 94(1): 21.
[30]. Ghanbari, B, Inc, M, Yusuf, A, Bayram, M. 2019. Exact optical solitons of Radhakrishnanâ “Kunduâ “Lakshmanan equation with Kerr law nonlinearity. Modern Physics Letters B; 33(06): 1950061.
[31]. Saglam Ozkan, Y. 2021. The generalized exponential rational function and Elzaki-Adomian decomposition method for the Heisenberg ferromagnetic spin chain equation. Modern Physics Letters B; 35(12): 2150200.
[32]. Kuo, CK, Gunay, B, Juan, CJ. 2023. The applications of symbolic computation to exact wave solutions of two HSI-like equations in (2+ 1)-dimensional. Frontiers in Physics; 11, 26.
[33]. Ghanbari, B, Osman, MS, Baleanu, D. 2019. Generalized exponential rational function method for extended Zakharov-Kuzetsov equation with conformable derivative. Modern Physics Letters A; 34(20): 1950155.
[34]. Ghanbari, B, Akgul, A. 2020. Abundant new analytical and approximate solutions to the generalized Schamel equation. Physica Scripta; 95(7): 075201.