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EXACT TRAVELING WAVE SOLUTIONS FOR THE NON-LINEAR COUPLE DRINFEL’D-SOKOLOV-WILSON (DSW) DYNAMICAL SYSTEM USING EXTENDED JACOBI ELLIPTIC FUNCTION EXPANSION METHOD

Yıl 2024, Cilt: 25 Sayı: 1, 141 - 155, 28.03.2024

Nisa Çelik

https://doi.org/10.18038/estubtda.1399247

Öz

The study of water waves is significant for researchers working in many branches of science. The behaviour of waves can be studied by observation or experimental means, but theoretically, mathematical modeling provides solutions to many problems in physics and engineering. Progress in this field is inevitable, with those who work in mathematics, physics, and engineering putting forth interdisciplinary studies.

Jacobi elliptic functions are valuable mathematical tools that can be applied to various aspects of mathematics, physics, and ocean engineering. In this study, traveling wave solutions of the general Drinfel'd-Sokolov-Wilson (DSW) system, introduced as a model of water waves, were obtained by using Jacobi elliptic functions and the wave dynamics were examined. The extended Jacobi elliptic function expansion method is an effective method for generating periodic solutions. It has been observed that the periodic solutions obtained by using Jacobi elliptic function expansions containing different Jacobi elliptic functions may be different and some new periodic solutions can be obtained. 3D simulations were made using MapleTM to see the behaviour of the solutions obtained for different appropriate values of the parameters. 2D simulations are presented for easy observation of wave motion. In addition, we transformed the one of the exact solutions found by the extended Jacobi elliptic function expansion method into the new solution under the symmetry transformation.

Anahtar Kelimeler

DSW system Exact solutions Extended Jacobi elliptic function expansion method

Kaynakça

  • [1] Andre EM, Costa DO. Fernandes AC. Junior JSS. A review on the modelling of subsea lifting operations, Ocean Engineering, 2023; 268: 113293.
  • [2] Sun H, Zong G, Cui J, Shi K. Fixed-time sliding mode output feedback tracking control for autonomous underwater vehicle with prescribed performance constraint. Ocean Engineering, 2022 ; 247: 110673.
  • [3] Wazwaz AM. Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Saint Xavier University, 2009; Chicago, IL 60655.
  • [4] Çelik N, Seadawy AR., Sağlam Y, Yaşar E. A model of solitary waves in a nonlinear elastic circular rod: Abundant different type exact solutions and conservation laws. Choas, Solitons and Fractals, 2021; 143: 1-19, 110486.
  • [5] Sait S. Extended Jacobi elliptic function solutions for general boussinesq systems. Revista Mexicana de Física, 2023; 69.2 Mar-Apr: 021401-1.
  • [6] Sait S, Altunay R. Abundant travelling wave solutions of 3+ 1 dimensional Boussinesq equation with dual dispersion. Revista Mexicana de Física, E 2022; 19 (2 Jul-Dec): 020203-1.
  • [7] Rezazadeh H, Davadi AG, Gholami D. Combined formal periodic wave-like and soliton-like solutions of the conformable Schrödinger-KdV equation using the G’/G-expansion technique. Results in Physics, 2023; 47-106352.
  • [8] San S, Yaşar E. On the Lie symmetry analysis, analytic series solutions, and conservation laws of the time fractional Belousov–Zhabotinskii system. Nonlinear Dynamics, 2022; 109(4): 2997-3008.
  • [9] Jhangeer A, Seadawy A.R, Ali F, Ahmed Abbirah. New complex waves of perturbed Shrödinger equation with Kerr law nonlinearity and Kundu-Mukherjee-Naskar equation. Result in Physics, 2020; 16-102816.
  • [10] Rehman UH, Ullah N, Imran MA. Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic. method. International Journal for Light and Electron Optics-Optik, 2021; 226: 1-15, 165378.
  • [11] Günhan AN, Yaşar E. A new (3+ 1) dimensional Hirota bilinear equation: Painlavé integrability, Lie symmetry analysis, and conservation laws. Journal of Taibah University for Science, 2022; 16(1): 1287-1297.
  • [12] Rabie WB, Ahmed HM, Mirzazadeh M, Akbulut A, Hashemi MS. Investigation of solitons and conservation laws in an inhomogeneous optical fiber through a generalized derivative nonlinear Schrödinger equation with quintic nonlinearity. Optical and Quantum Electronics, 2023; 55(9): 825.
  • [13] Drinfel’d VG, Sokolov VV. Equations of Korteweg–de Vries type, and simple Lie algebras Sov. Math. Dokl., 1981; 23: 457–462.
  • [14] Wilson G. The affine Lie algebra C (1) 2 and an equation of Hirota and Satsuma. Physics Letters A, 1982; 89.7: 332-334.
  • [15] Shen S, Ding X, Zhang R, Hu X. Intereaction solutions to the (1+1)-dimensional generalized Drinfel’d-Sokolov-Wilson equation. Modern Physics Letters B, 2019; 33(27) :1950329.
  • [16] Bashar MdH, Yiasir SMA, Islam SMR, Rahman MM. Wave solutions of the couple Drinfel’d-Sokolov-Wilson equation: New wave solutions and free parameters effect. Journal of Ocean Engineering and Science, 2022.
  • [17] Khan K, Akbar M A, Koppelaar H. Study of coupled nonlinear partial differential equations for finding exact analytical solutions. Royal Society open science, 2015: 2(7); 140406.
  • [18] Ren B, Lin J, Lou ZM. Consistent Riccati expansion and rational solutions of the Drinfel’d Sokolov-Wilson equation. Applied Mathematics Letter, 2020: 105; 106326.
  • [19] Emad HM, Zahran, Mostafa MA Khater. Exact Traveling Wave Solutions for the System of Shallow Water Wavw Equations and Modified Liouville Equation Using Extanded Jacobian Elliptic Function Expansion Method. American Journal of Computational Mathematics, 2014: 4; 455-463.
  • [20] Çelik N. Nonlinear models in Ocean Engineering: Exact solutions, 3D and 2D simulations of the General Drinfel’d-Sokolov-Wilson system with Jacobi Elliptic Functions. MAS 18th International European Conference on Mathematics, Engineering, Natural&Medical Sciences, 2023; 112-119.
  • [21] Zhang Y, Zhao Z. Lie symmetry analysis, Lie Backlund symmetries, explicit solutions, and conservation laws of Drinfeld-Sokolov-Wilson system. Boundry Value Problems, 2017: 154; 2-7.

EXACT TRAVELING WAVE SOLUTIONS FOR THE NON-LINEAR COUPLE DRINFEL’D-SOKOLOV-WILSON (DSW) DYNAMICAL SYSTEM USING EXTENDED JACOBI ELLIPTIC FUNCTION EXPANSION METHOD

Yıl 2024, Cilt: 25 Sayı: 1, 141 - 155, 28.03.2024

Nisa Çelik

https://doi.org/10.18038/estubtda.1399247

Öz

The study of water waves is significant for researchers working in many branches of science. The behaviour of waves can be studied by observation or experimental means, but theoretically, mathematical modeling provides solutions to many problems in physics and engineering. Progress in this field is inevitable, with those who work in mathematics, physics, and engineering putting forth interdisciplinary studies.

Jacobi elliptic functions are valuable mathematical tools that can be applied to various aspects of mathematics, physics, and ocean engineering. In this study, traveling wave solutions of the general Drinfel'd-Sokolov-Wilson (DSW) system, introduced as a model of water waves, were obtained by using Jacobi elliptic functions and the wave dynamics were examined. The extended Jacobi elliptic function expansion method is an effective method for generating periodic solutions. It has been observed that the periodic solutions obtained by using Jacobi elliptic function expansions containing different Jacobi elliptic functions may be different and some new periodic solutions can be obtained. 3D simulations were made using MapleTM to see the behaviour of the solutions obtained for different appropriate values of the parameters. 2D simulations are presented for easy observation of wave motion. In addition, we transformed the one of the exact solutions found by the extended Jacobi elliptic function expansion method into the new solution under the symmetry transformation.

Anahtar Kelimeler

DSW system Exact solutions Extended Jacobi elliptic function expansion method

Kaynakça

  • [1] Andre EM, Costa DO. Fernandes AC. Junior JSS. A review on the modelling of subsea lifting operations, Ocean Engineering, 2023; 268: 113293.
  • [2] Sun H, Zong G, Cui J, Shi K. Fixed-time sliding mode output feedback tracking control for autonomous underwater vehicle with prescribed performance constraint. Ocean Engineering, 2022 ; 247: 110673.
  • [3] Wazwaz AM. Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science. Saint Xavier University, 2009; Chicago, IL 60655.
  • [4] Çelik N, Seadawy AR., Sağlam Y, Yaşar E. A model of solitary waves in a nonlinear elastic circular rod: Abundant different type exact solutions and conservation laws. Choas, Solitons and Fractals, 2021; 143: 1-19, 110486.
  • [5] Sait S. Extended Jacobi elliptic function solutions for general boussinesq systems. Revista Mexicana de Física, 2023; 69.2 Mar-Apr: 021401-1.
  • [6] Sait S, Altunay R. Abundant travelling wave solutions of 3+ 1 dimensional Boussinesq equation with dual dispersion. Revista Mexicana de Física, E 2022; 19 (2 Jul-Dec): 020203-1.
  • [7] Rezazadeh H, Davadi AG, Gholami D. Combined formal periodic wave-like and soliton-like solutions of the conformable Schrödinger-KdV equation using the G’/G-expansion technique. Results in Physics, 2023; 47-106352.
  • [8] San S, Yaşar E. On the Lie symmetry analysis, analytic series solutions, and conservation laws of the time fractional Belousov–Zhabotinskii system. Nonlinear Dynamics, 2022; 109(4): 2997-3008.
  • [9] Jhangeer A, Seadawy A.R, Ali F, Ahmed Abbirah. New complex waves of perturbed Shrödinger equation with Kerr law nonlinearity and Kundu-Mukherjee-Naskar equation. Result in Physics, 2020; 16-102816.
  • [10] Rehman UH, Ullah N, Imran MA. Optical solitons of Biswas-Arshed equation in birefringent fibers using extended direct algebraic. method. International Journal for Light and Electron Optics-Optik, 2021; 226: 1-15, 165378.
  • [11] Günhan AN, Yaşar E. A new (3+ 1) dimensional Hirota bilinear equation: Painlavé integrability, Lie symmetry analysis, and conservation laws. Journal of Taibah University for Science, 2022; 16(1): 1287-1297.
  • [12] Rabie WB, Ahmed HM, Mirzazadeh M, Akbulut A, Hashemi MS. Investigation of solitons and conservation laws in an inhomogeneous optical fiber through a generalized derivative nonlinear Schrödinger equation with quintic nonlinearity. Optical and Quantum Electronics, 2023; 55(9): 825.
  • [13] Drinfel’d VG, Sokolov VV. Equations of Korteweg–de Vries type, and simple Lie algebras Sov. Math. Dokl., 1981; 23: 457–462.
  • [14] Wilson G. The affine Lie algebra C (1) 2 and an equation of Hirota and Satsuma. Physics Letters A, 1982; 89.7: 332-334.
  • [15] Shen S, Ding X, Zhang R, Hu X. Intereaction solutions to the (1+1)-dimensional generalized Drinfel’d-Sokolov-Wilson equation. Modern Physics Letters B, 2019; 33(27) :1950329.
  • [16] Bashar MdH, Yiasir SMA, Islam SMR, Rahman MM. Wave solutions of the couple Drinfel’d-Sokolov-Wilson equation: New wave solutions and free parameters effect. Journal of Ocean Engineering and Science, 2022.
  • [17] Khan K, Akbar M A, Koppelaar H. Study of coupled nonlinear partial differential equations for finding exact analytical solutions. Royal Society open science, 2015: 2(7); 140406.
  • [18] Ren B, Lin J, Lou ZM. Consistent Riccati expansion and rational solutions of the Drinfel’d Sokolov-Wilson equation. Applied Mathematics Letter, 2020: 105; 106326.
  • [19] Emad HM, Zahran, Mostafa MA Khater. Exact Traveling Wave Solutions for the System of Shallow Water Wavw Equations and Modified Liouville Equation Using Extanded Jacobian Elliptic Function Expansion Method. American Journal of Computational Mathematics, 2014: 4; 455-463.
  • [20] Çelik N. Nonlinear models in Ocean Engineering: Exact solutions, 3D and 2D simulations of the General Drinfel’d-Sokolov-Wilson system with Jacobi Elliptic Functions. MAS 18th International European Conference on Mathematics, Engineering, Natural&Medical Sciences, 2023; 112-119.
  • [21] Zhang Y, Zhao Z. Lie symmetry analysis, Lie Backlund symmetries, explicit solutions, and conservation laws of Drinfeld-Sokolov-Wilson system. Boundry Value Problems, 2017: 154; 2-7.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Uygulamalı Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Nisa Çelik 0000-0003-1209-991X

Yayımlanma Tarihi 28 Mart 2024
Gönderilme Tarihi 2 Aralık 2023
Kabul Tarihi 20 Mart 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 25 Sayı: 1

Kaynak Göster

AMA Çelik N. EXACT TRAVELING WAVE SOLUTIONS FOR THE NON-LINEAR COUPLE DRINFEL’D-SOKOLOV-WILSON (DSW) DYNAMICAL SYSTEM USING EXTENDED JACOBI ELLIPTIC FUNCTION EXPANSION METHOD. Eskişehir Technical University Journal of Science and Technology A - Applied Sciences and Engineering. Mart 2024;25(1):141-155. doi:10.18038/estubtda.1399247