# American Institute of Mathematical Sciences

## On the new wave solutions to the Wu-Zhang system by expansion methods

 Department of Mathematics and Science Education, Faculty of Education, Ordu University, Turkey

* Corresponding author: tolgaakturkk@gmail.com

Received  April 2019 Revised  May 2019 Published  October 2019

In this article, Some solitary wave solutions to the Wu-Zhang system are obtained by using the modified expansion function method and the sine-Gordon expansion method. We get solitary wave solutions with the hyperbolic function structures and trigonometric to this nonlinear model. Two and three-dimensional graphics are plotted by choosing the appropriate parameters values. All the solutions obtained satisfy the system of equations in the model. All mathematical calculations in this work are done with the help of Wolfram Mathematica software.

Citation: Tolga Aktürk. On the new wave solutions to the Wu-Zhang system by expansion methods. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020125
##### References:
 [1] M. A. Abdou, The extended tanh method and its applications for solving nonlinear physical models, Applied Mathematics and Computation, 190 (2007), 988–996, https://www.sciencedirect.com/science/article/pii/S0096300307000756. doi: 10.1016/j.amc.2007.01.070.  Google Scholar [2] H. M. Baskonus and H. Bulut, On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method, Waves in Random and Complex Media, 25 (2015), 720–728. Google Scholar [3] H. M. Baskonus, H. Bulut and A. Atangana, On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod, Smart Materials and Structures, 25 (2016), 035022, . Google Scholar [4] H. M. Baskonus, H. Bulut and T. A. Sulaiman, Investigation of various travelling wave so lutions to the extended (2+1)-dimensional quantum ZK equation, The European Physical Journal Plus, 132 (2017), 482. Google Scholar [5] H. M. Baskonus, T. A. Sulaiman and H. Bulut, On the novel wave behaviors to the coupled nonlinear Maccari's system with complex structure, Optik, 131 (2017), 1036–1043. Google Scholar [6] H. M. Baskonus and H. Bulut, Exponential prototype structures for (2+1)-dimensional Boiti Leon-Pempinelli systems in mathematical physics, Waves in Random and Complex Media, 26 (2016), 189–196. Google Scholar [7] H. Bulut and T. Akturk, Traveling wave solutions of the (N+1)-dimensional sine-cosine Gordon equation, AIP Conference Proceedings, 1637 (2014), 145–149. Google Scholar [8] H. Bulut, T. A. Sulaiman and H. M. Baskonus, On the new soliton and optical wave structures to some nonlinear evolution equations, The European Physical Journal Plus, 132 (2017), 459. Google Scholar [9] H. Bulut, T. A. Sulaiman and T. Yazgan, Novel hyperbolic behaviors to some important models arising in quantum science, Optical and Quantum Electronics, 49 (2017), 349. Google Scholar [10] H. Bulut, T. A. Sulaiman, H. M. Baskonus and A. A. Sandulyak, New solitary and optical wave structures to the (1+1)-dimensional combined KdV-mKdV equation, Optik, 135 (2017), 327–336. Google Scholar [11] H. Bulut, T. A. Sulaiman and B. Demirdag, Dynamics of soliton solutions in the chiral nonlinear Schrödinger equations, Nonlinear Dynamics, 91 (2018), 1985–1991. Google Scholar [12] Y. Chen and Z. Y. Yan, New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method, Chaos, Solitons & Fractals, 26 (2005), 399–406, https://www.sciencedirect.com/science/article/pii/S0960077905000755. doi: 10.1016/j.chaos.2005.01.004.  Google Scholar [13] O. Cornejo-Pérez and H. C. Rosu, Nonlinear second order Ode's: Factorizations and particular solutions, Progress of Theoretical Physics, 114 (2005), 533–538, https://academic.oup.com/ptp/article-abstract/114/3/533/1823840. doi: 10.1143/PTP.114.533.  Google Scholar [14] M. T. Darvishi and M. Najafi, Some complexiton type solutions of the (3+1)-dimensional Jimbo-Miwa equation, World Academyof Science, Engineering and Technology, 55 (2011), 1097–1099. Google Scholar [15] M. Eslami and H. Rezazadeh, The first integral method for Wu–Zhang system with con formable time-fractional derivative, Calcolo, 53 (2016), 475–485. Google Scholar [16] Z. Hammouch and T. Mekkaoui, Travelling-wave solutions of the generalized Zakharov equa tion with time-space fractional derivatives, Journal: MESA, 5 (2014), 489–498. Google Scholar [17] Z. Hammouch, T. Mekkaoui and P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii Schiff equation in (2+1) dimensions with time-fractional conformable derivative, The Euro pean Physical Journal Plus, 133 (2018), 248. Google Scholar [18] J.-H. He and X.-H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30 (2006), 700–708, https://www.sciencedirect.com/science/article/pii/S0960077906002293. doi: 10.1016/j.chaos.2006.03.020.  Google Scholar [19] W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. van Immerzeele and A. Meerpoel, Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, Journal of Physics A: Mathematical and General, 19 (1986), 607-628.  doi: 10.1088/0305-4470/19/5/016.  Google Scholar [20] O. A. Ilhan, H. Bulut, T. A. Sulaiman and H. M. Baskonus, Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd-Bullough-Mikhailov equation, Indian Journal of Physics, 92 (2018), 999–1007. Google Scholar [21] H. Jafari, R. Soltani, C. M. Khalique and D. Baleanu, Exact solutions of two nonlinear partial differential equations by using the first integral method, Boundary Value Problems, 2013 (2013), 9 pp. doi: 10.1186/1687-2770-2013-117.  Google Scholar [22] K. Khan and M. Ali Akbar, Application of exp-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation, World Applied Sciences Journal, 24 (2013), 1373–1377. Google Scholar [23] S. Kumar, K. Singh and R. K. Gupta, Coupled Higgs field equation and Hamiltonian ampli tude equation: Lie classical approach and (G$\mathrm{\prime}$/G)-expansion method, Pramana, 79 (2012), 41–60. Google Scholar [24] S. K. Liu, Z. T. Fu, S. D. Liu and Q. zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69-74.  doi: 10.1016/S0375-9601(01)00580-1.  Google Scholar [25] D. C. Lu, B. J. Hong and L. X. Tian, Backlund transformation and n-soliton-like solutions to the combined KdV-Burgers equation with variable coefficients, International Journal of Nonlinear Science, 2 (2006), 3–10.  Google Scholar [26] W.-X. Ma, T. W. Huang and Y. Zhang, A multiple exp-function method for nonlinear differ ential equations and its application, Physica Scripta, 82 (2010), 065003.  Google Scholar [27] A. J. Mohamad Jawad, M. D. Petković and A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217 (2010), 869–877, https://www.sciencedirect.com/science/article/pii/S0096300310007150. doi: 10.1016/j.amc.2010.06.030.  Google Scholar [28] H. Naher and F. A. Abdullah, New generalized and improved (G$\mathrm{\prime}$/G)-expansion method for nonlinear evolution equations in mathematical physics, Journal of the Egyptian Mathematical Society, 22 (2014), 390–395, https://www.sciencedirect.com/science/article/pii/S1110256X13001405. doi: 10.1016/j.joems.2013.11.008.  Google Scholar [29] OŠ, T. et al. A Short Review on Analytical methods for fractional equations with he's fractional derivative, Thermal Science, 21 (2017), 1567–1574. Google Scholar [30] A. R. Seadawy, Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves, The European Physical Journal Plus, 132 (2017), 29. Google Scholar [31] T. A. Sulaiman, T. Aktürk, H. Bulut and H. M. Baskonus, Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 32 (2018), 1093–1105. Google Scholar [32] M. L. Wang, M. B. Zhou and Z. B. Lin, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216 (1996), 67–75. Google Scholar [33] A. Yokus, T. A. Sulaiman and H. Bulut, On the analytical and numerical solutions of the Benjamin-Bona-Mahony equation, Optical and Quantum Electronics, 50 (2018), 31. Google Scholar [34] A. Yokus, H. M. Baskonus, T. A. Sulaiman and H. Bulut, Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34 (2018), 211–227. Google Scholar [35] H. Zakia and T. Mekkaoui, Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1 (2012), 206–212. Google Scholar [36] E. M. E. Zayed and S. A. Hoda lbrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chinese Physics Letters, 29 (2012), 060201. Google Scholar [37] Z. Y. Zhang, Jacobi elliptic function expansion method for the modified Korteweg-de Vries Zakharov-Kuznetsov and the Hirota equations, Romanian Journal of Physics, 60 (2015), 1384–1394. Google Scholar

show all references

##### References:
 [1] M. A. Abdou, The extended tanh method and its applications for solving nonlinear physical models, Applied Mathematics and Computation, 190 (2007), 988–996, https://www.sciencedirect.com/science/article/pii/S0096300307000756. doi: 10.1016/j.amc.2007.01.070.  Google Scholar [2] H. M. Baskonus and H. Bulut, On the complex structures of Kundu-Eckhaus equation via improved Bernoulli sub-equation function method, Waves in Random and Complex Media, 25 (2015), 720–728. Google Scholar [3] H. M. Baskonus, H. Bulut and A. Atangana, On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod, Smart Materials and Structures, 25 (2016), 035022, . Google Scholar [4] H. M. Baskonus, H. Bulut and T. A. Sulaiman, Investigation of various travelling wave so lutions to the extended (2+1)-dimensional quantum ZK equation, The European Physical Journal Plus, 132 (2017), 482. Google Scholar [5] H. M. Baskonus, T. A. Sulaiman and H. Bulut, On the novel wave behaviors to the coupled nonlinear Maccari's system with complex structure, Optik, 131 (2017), 1036–1043. Google Scholar [6] H. M. Baskonus and H. Bulut, Exponential prototype structures for (2+1)-dimensional Boiti Leon-Pempinelli systems in mathematical physics, Waves in Random and Complex Media, 26 (2016), 189–196. Google Scholar [7] H. Bulut and T. Akturk, Traveling wave solutions of the (N+1)-dimensional sine-cosine Gordon equation, AIP Conference Proceedings, 1637 (2014), 145–149. Google Scholar [8] H. Bulut, T. A. Sulaiman and H. M. Baskonus, On the new soliton and optical wave structures to some nonlinear evolution equations, The European Physical Journal Plus, 132 (2017), 459. Google Scholar [9] H. Bulut, T. A. Sulaiman and T. Yazgan, Novel hyperbolic behaviors to some important models arising in quantum science, Optical and Quantum Electronics, 49 (2017), 349. Google Scholar [10] H. Bulut, T. A. Sulaiman, H. M. Baskonus and A. A. Sandulyak, New solitary and optical wave structures to the (1+1)-dimensional combined KdV-mKdV equation, Optik, 135 (2017), 327–336. Google Scholar [11] H. Bulut, T. A. Sulaiman and B. Demirdag, Dynamics of soliton solutions in the chiral nonlinear Schrödinger equations, Nonlinear Dynamics, 91 (2018), 1985–1991. Google Scholar [12] Y. Chen and Z. Y. Yan, New exact solutions of (2+1)-dimensional Gardner equation via the new sine-Gordon equation expansion method, Chaos, Solitons & Fractals, 26 (2005), 399–406, https://www.sciencedirect.com/science/article/pii/S0960077905000755. doi: 10.1016/j.chaos.2005.01.004.  Google Scholar [13] O. Cornejo-Pérez and H. C. Rosu, Nonlinear second order Ode's: Factorizations and particular solutions, Progress of Theoretical Physics, 114 (2005), 533–538, https://academic.oup.com/ptp/article-abstract/114/3/533/1823840. doi: 10.1143/PTP.114.533.  Google Scholar [14] M. T. Darvishi and M. Najafi, Some complexiton type solutions of the (3+1)-dimensional Jimbo-Miwa equation, World Academyof Science, Engineering and Technology, 55 (2011), 1097–1099. Google Scholar [15] M. Eslami and H. Rezazadeh, The first integral method for Wu–Zhang system with con formable time-fractional derivative, Calcolo, 53 (2016), 475–485. Google Scholar [16] Z. Hammouch and T. Mekkaoui, Travelling-wave solutions of the generalized Zakharov equa tion with time-space fractional derivatives, Journal: MESA, 5 (2014), 489–498. Google Scholar [17] Z. Hammouch, T. Mekkaoui and P. Agarwal, Optical solitons for the Calogero-Bogoyavlenskii Schiff equation in (2+1) dimensions with time-fractional conformable derivative, The Euro pean Physical Journal Plus, 133 (2018), 248. Google Scholar [18] J.-H. He and X.-H. Wu, Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30 (2006), 700–708, https://www.sciencedirect.com/science/article/pii/S0960077906002293. doi: 10.1016/j.chaos.2006.03.020.  Google Scholar [19] W. Hereman, P. P. Banerjee, A. Korpel, G. Assanto, A. van Immerzeele and A. Meerpoel, Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method, Journal of Physics A: Mathematical and General, 19 (1986), 607-628.  doi: 10.1088/0305-4470/19/5/016.  Google Scholar [20] O. A. Ilhan, H. Bulut, T. A. Sulaiman and H. M. Baskonus, Dynamic of solitary wave solutions in some nonlinear pseudoparabolic models and Dodd-Bullough-Mikhailov equation, Indian Journal of Physics, 92 (2018), 999–1007. Google Scholar [21] H. Jafari, R. Soltani, C. M. Khalique and D. Baleanu, Exact solutions of two nonlinear partial differential equations by using the first integral method, Boundary Value Problems, 2013 (2013), 9 pp. doi: 10.1186/1687-2770-2013-117.  Google Scholar [22] K. Khan and M. Ali Akbar, Application of exp-expansion method to find the exact solutions of modified Benjamin-Bona-Mahony equation, World Applied Sciences Journal, 24 (2013), 1373–1377. Google Scholar [23] S. Kumar, K. Singh and R. K. Gupta, Coupled Higgs field equation and Hamiltonian ampli tude equation: Lie classical approach and (G$\mathrm{\prime}$/G)-expansion method, Pramana, 79 (2012), 41–60. Google Scholar [24] S. K. Liu, Z. T. Fu, S. D. Liu and Q. zhao, Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations, Phys. Lett. A, 289 (2001), 69-74.  doi: 10.1016/S0375-9601(01)00580-1.  Google Scholar [25] D. C. Lu, B. J. Hong and L. X. Tian, Backlund transformation and n-soliton-like solutions to the combined KdV-Burgers equation with variable coefficients, International Journal of Nonlinear Science, 2 (2006), 3–10.  Google Scholar [26] W.-X. Ma, T. W. Huang and Y. Zhang, A multiple exp-function method for nonlinear differ ential equations and its application, Physica Scripta, 82 (2010), 065003.  Google Scholar [27] A. J. Mohamad Jawad, M. D. Petković and A. Biswas, Modified simple equation method for nonlinear evolution equations, Applied Mathematics and Computation, 217 (2010), 869–877, https://www.sciencedirect.com/science/article/pii/S0096300310007150. doi: 10.1016/j.amc.2010.06.030.  Google Scholar [28] H. Naher and F. A. Abdullah, New generalized and improved (G$\mathrm{\prime}$/G)-expansion method for nonlinear evolution equations in mathematical physics, Journal of the Egyptian Mathematical Society, 22 (2014), 390–395, https://www.sciencedirect.com/science/article/pii/S1110256X13001405. doi: 10.1016/j.joems.2013.11.008.  Google Scholar [29] OŠ, T. et al. A Short Review on Analytical methods for fractional equations with he's fractional derivative, Thermal Science, 21 (2017), 1567–1574. Google Scholar [30] A. R. Seadawy, Travelling-wave solutions of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves, The European Physical Journal Plus, 132 (2017), 29. Google Scholar [31] T. A. Sulaiman, T. Aktürk, H. Bulut and H. M. Baskonus, Investigation of various soliton solutions to the Heisenberg ferromagnetic spin chain equation, Journal of Electromagnetic Waves and Applications, 32 (2018), 1093–1105. Google Scholar [32] M. L. Wang, M. B. Zhou and Z. B. Lin, Application of a homogeneous balance method to exact solutions of nonlinear equations in mathematical physics, Physics Letters A, 216 (1996), 67–75. Google Scholar [33] A. Yokus, T. A. Sulaiman and H. Bulut, On the analytical and numerical solutions of the Benjamin-Bona-Mahony equation, Optical and Quantum Electronics, 50 (2018), 31. Google Scholar [34] A. Yokus, H. M. Baskonus, T. A. Sulaiman and H. Bulut, Numerical simulation and solutions of the two-component second order KdV evolutionary system, Numerical Methods for Partial Differential Equations, 34 (2018), 211–227. Google Scholar [35] H. Zakia and T. Mekkaoui, Travelling-wave solutions for some fractional partial differential equation by means of generalized trigonometry functions, International Journal of Applied Mathematical Research, 1 (2012), 206–212. Google Scholar [36] E. M. E. Zayed and S. A. Hoda lbrahim, Exact solutions of nonlinear evolution equations in mathematical physics using the modified simple equation method, Chinese Physics Letters, 29 (2012), 060201. Google Scholar [37] Z. Y. Zhang, Jacobi elliptic function expansion method for the modified Korteweg-de Vries Zakharov-Kuznetsov and the Hirota equations, Romanian Journal of Physics, 60 (2015), 1384–1394. Google Scholar
The 3-dimensional, density and 2-dimensional graphs of Eq. (30) under the values $k=1,\, \, \lambda =3,E=0.75\, ,c=1,\, \, $$B_{0} =0.35 and t=1 for the 2D graph The 3-dimensional, density and 2-dimensional graphs of Eq. (32) under the values k=1,\, \, \lambda =3,\, \, c=1\, ,B_{0} =0.35$$ ,E=0.75$ and $t=1$ for the 2D graph
The 3-dimensional, density and 2-dimensional graphs of Eq. (34) under the values $k=1,\, \, \lambda =3,\, \, c=1, $$E=0.75\,$$ \, B_{0} =0.35$and $t=1$ for the 2D graph
The 3-dimensional, density and 2-dimensional graphs of Eq. (36) under the values $k=1,\, \, c=1,\lambda =3,\, $$E=0.75\, ,$$ s=1$ and$B_{0} =0.35, $$t=1 for the 2D graph The 3-dimensional, density and 2-dimensional graphs of Eq. (38) under the values k=1,\, \, c=1,\lambda =3\, ,\, \, s=1,$$ B_{0} =0.35,E=0.75$and $t=1$ for the 2D graph
The 3-dimensional, density and 2-dimensional graphs of Eq. (40) under the values $k=1,\, \, c=1,\lambda =3\, ,\, \, s=1,$$B_{0} =0.35,E=0.75$ and $t=1$ for the 2D graph
The 3-dimensional, density and 2-dimensional graphs of the imaginary and real part of the Eq.(43)
The 3-dimensional, density and 2-dimensional graphs of the equation (45)
The 3-dimensional, density and 2-dimensional graphs of Eq.(47)
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