# American Institute of Mathematical Sciences

October  2020, 13(10): 2927-2939. doi: 10.3934/dcdss.2020214

## Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation

 1 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, Shandong 266590, China 2 African Institute for Mathematical Sciences, Muizenberg, Cape Town, South Africa 3 Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China 4 College of mechanical and automotive engineering, Zhejiang University of Water Resources and Electric Power, Hangzhou, Zhejiang 310018, China 5 International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho, 2735, South Africa

* Corresponding author: Lijun Zhang

Received  November 2018 Revised  July 2019 Published  October 2020 Early access  December 2019

Fund Project: This work is supported by NSF grant No. 11672270 and No.11872335

The bounded traveling wave solutions of the Zakharov-Rubenchik equation are investigated by using the method of dynamical system theorems in this paper. After suitable transformations we find that the traveling wave equations of the Zakharov-Rubenchik equation are fully determined by a second-order singular ordinary differential equation (ODE) with three real coefficients which can be arbitrary constants. We derive abundant exact bounded periodic and solitary wave solutions of the Zakharov-Rubenchik equation via studying the bifurcations and exact solutions of the derived ODE.

Citation: Lijun Zhang, Peiying Yuan, Jingli Fu, Chaudry Masood Khalique. Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2927-2939. doi: 10.3934/dcdss.2020214
##### References:
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Khalique, The effects of the singular lines on the traveling wave solutions of modified dispersive water wave equations, Nonlinear Anal. Real World Appl., 47 (2019), 236-250.  doi: 10.1016/j.nonrwa.2018.10.012. [7] J. He and X. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700-708.  doi: 10.1016/j.chaos.2006.03.020. [8] C. He, Y. Tang and J. Ma, New interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation, Compu. Math. Appl., 76 (2018), 2141-2147.  doi: 10.1016/j.camwa.2018.08.012. [9] M. Javidi and A. Golbabai, Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Soliton Fractals, 36 (2008), 309-313.  doi: 10.1016/j.chaos.2006.06.088. [10] F. Li and G. Du, General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback, J. Appl. Anal. Compu., 8 (2018), 390-401. [11] X. Li and Q. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.  doi: 10.1016/j.geomphys.2017.07.010. [12] F. Li and Q. Gao, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Compu., 274 (2016), 383-392.  doi: 10.1016/j.amc.2015.11.018. [13] J. Li, Singular Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. [14] J. Li, Geometric properties and exact traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation, J. Nonlinear Modeling Analysis, 1 (2019), 1-10. [15] F. Linares and C. Matheus, Well posedness for the 1D Zakharov-Rubenchik system, Adv. Differential Equations, 14 (2009), 261-288. [16] T. Liu and H. Dong, The prolongation structure of the modified nonlinear Schrödinger equation and its initial-boundary value problem on the half line via the Riemann-Hilbert approach, Mathematics, 7 (2019), 170 pp. doi: 10.3390/math7020170. [17] Y. Liu, H. Dong and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys., 9 (2019), 465-481.  doi: 10.1007/s13324-018-0209-9. [18] C. Lu, L. Xie and H. Yang, Analysis of Lie symmetries with conservation laws and solutions for the generalized (3 + 1)-dimensional time fractional Camassa-Holm-Kadomtsev-Petviashvili equation, Compu. Math. Appl., 77 (2019), 3154-3171.  doi: 10.1016/j.camwa.2019.01.022. [19] Y. Ren, M. Tao, H. Dong and H. Yang, Analytical research of (3+1)-dimensional Rossby waves with dissipation effect in cylindrical coordinate based on Lie symmetry approach, Adv. Difference Equ., 13 (2019), 9 pp. doi: 10.1186/s13662-019-1952-4. [20] W. G. Rui, Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2667-2678.  doi: 10.1016/j.cnsns.2013.01.020. [21] F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation, Phys. D, 175 (2003), 220-240.  doi: 10.1016/S0167-2789(02)00722-4. [22] F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik Equation, Reports Math. Phys., 61 (2008), 13-27.  doi: 10.1016/S0034-4877(08)00006-2. [23] G. Ponce and J. C. Saut, Well-posedness for the Benney-Roskes/Zakharov-Rubenchik system, Disc. Contin. Dyn. Syst., 13 (2005), 811-825.  doi: 10.3934/dcds.2005.13.811. [24] Y. Wang, C. Dai, L. Wu and J. Zhang, Exact and numerical solitary wave solutions of generalized Zakharov equation by the Adomian decomposition method, Chaos Soliton Fractals, 32 (2007), 1208-1214.  doi: 10.1016/j.chaos.2005.11.071. [25] V. E. Zakharov, Collapse of Langmuir Waves, Soviet Physics-JETP, 35 (1972), 908-914. [26] Y. Zhang, H. H. Dong, X. E. Zhang and H. W. Yang, Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-like equation, Compu. Math. Appl., 73 (2017), 246-252.  doi: 10.1016/j.camwa.2016.11.009. [27] L. Zhang and C. M. Khalique, Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759-772.  doi: 10.3934/dcdss.2018048. [28] L. Zhang, Y. Wang, C. M. Khlique and Y. Bai, Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation, J. Appl. Anal. Compu., 8 (2018), 1938-1958. [29] J. Zhang, L. Zhang and Y. Bai, Stability and bifurcation analysis on a predator prey system with the weak allee effect, Mathematics, 7 (2019), 1-15.  doi: 10.3390/math7050432. [30] H. Zhao and W. Ma, Mixed lumpkink solutions to the KP equation, Compu. Math. Appl., 74 (2017), 1399-1405.  doi: 10.1016/j.camwa.2017.06.034. [31] Q. L. Zhao and X. Y. Li, A Bargmann system and the involutive solutions associated with a new 4-Order Lattice hierarchy, Anal. Math Phys., 6 (2016), 237-254.  doi: 10.1007/s13324-015-0116-2. [32] X. X. Zheng, Y. D. Shang and X. M. Peng, Orbital stability of solitary waves of the coupled Klein-Gordon-Zakharov equations, Math. Meth. Appl. Sci., 40 (2017), 2623-2633.  doi: 10.1002/mma.4187.

show all references

##### References:
 [1] S. Abbasbandy, E. Babolian and M. Ashtiani, Numerical solution of the generalized Zakharov equation by homotopy analysis method, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 4114-4121.  doi: 10.1016/j.cnsns.2009.03.001. [2] P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, New York-Berlin, 1971. [3] H. Ding, C. W. Lim and L. Q. Chen, Nonlinear vibration of a traveling belt with non-homogeneous boundaries, J Sound Vib., 424 (2018), 78-93.  doi: 10.1016/j.jsv.2018.03.010. [4] J. Gu, Y. Zhang and H. Dong, Dynamic behaviors of interaction solutions of (3+1)-dimensional Shallow Water wave equation, Comp. Math. Appl., 76 (2018), 1408-1419.  doi: 10.1016/j.camwa.2018.06.034. [5] B. Guo, J. Zhang and X. Pu, On the existence and uniqueness of smooth solution for a generalized Zakharov equation, J. Math. Anal. Appl., 365 (2010), 238-253.  doi: 10.1016/j.jmaa.2009.10.045. [6] M. Han, L. Zhang, Y. Wang and C. M. Khalique, The effects of the singular lines on the traveling wave solutions of modified dispersive water wave equations, Nonlinear Anal. Real World Appl., 47 (2019), 236-250.  doi: 10.1016/j.nonrwa.2018.10.012. [7] J. He and X. Wu, Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700-708.  doi: 10.1016/j.chaos.2006.03.020. [8] C. He, Y. Tang and J. Ma, New interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation, Compu. Math. Appl., 76 (2018), 2141-2147.  doi: 10.1016/j.camwa.2018.08.012. [9] M. Javidi and A. Golbabai, Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Soliton Fractals, 36 (2008), 309-313.  doi: 10.1016/j.chaos.2006.06.088. [10] F. Li and G. Du, General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback, J. Appl. Anal. Compu., 8 (2018), 390-401. [11] X. Li and Q. Zhao, A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.  doi: 10.1016/j.geomphys.2017.07.010. [12] F. Li and Q. Gao, Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Compu., 274 (2016), 383-392.  doi: 10.1016/j.amc.2015.11.018. [13] J. Li, Singular Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. [14] J. Li, Geometric properties and exact traveling wave solutions for the generalized Burger-Fisher equation and the Sharma-Tasso-Olver equation, J. Nonlinear Modeling Analysis, 1 (2019), 1-10. [15] F. Linares and C. Matheus, Well posedness for the 1D Zakharov-Rubenchik system, Adv. Differential Equations, 14 (2009), 261-288. [16] T. Liu and H. Dong, The prolongation structure of the modified nonlinear Schrödinger equation and its initial-boundary value problem on the half line via the Riemann-Hilbert approach, Mathematics, 7 (2019), 170 pp. doi: 10.3390/math7020170. [17] Y. Liu, H. Dong and Y. Zhang, Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys., 9 (2019), 465-481.  doi: 10.1007/s13324-018-0209-9. [18] C. Lu, L. Xie and H. Yang, Analysis of Lie symmetries with conservation laws and solutions for the generalized (3 + 1)-dimensional time fractional Camassa-Holm-Kadomtsev-Petviashvili equation, Compu. Math. Appl., 77 (2019), 3154-3171.  doi: 10.1016/j.camwa.2019.01.022. [19] Y. Ren, M. Tao, H. Dong and H. Yang, Analytical research of (3+1)-dimensional Rossby waves with dissipation effect in cylindrical coordinate based on Lie symmetry approach, Adv. Difference Equ., 13 (2019), 9 pp. doi: 10.1186/s13662-019-1952-4. [20] W. G. Rui, Different kinds of exact solutions with two-loop character of the two-component short pulse equations of the first kind, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 2667-2678.  doi: 10.1016/j.cnsns.2013.01.020. [21] F. Oliveira, Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation, Phys. D, 175 (2003), 220-240.  doi: 10.1016/S0167-2789(02)00722-4. [22] F. Oliveira, Adiabatic limit of the Zakharov-Rubenchik Equation, Reports Math. Phys., 61 (2008), 13-27.  doi: 10.1016/S0034-4877(08)00006-2. [23] G. Ponce and J. C. Saut, Well-posedness for the Benney-Roskes/Zakharov-Rubenchik system, Disc. Contin. Dyn. Syst., 13 (2005), 811-825.  doi: 10.3934/dcds.2005.13.811. [24] Y. Wang, C. Dai, L. Wu and J. Zhang, Exact and numerical solitary wave solutions of generalized Zakharov equation by the Adomian decomposition method, Chaos Soliton Fractals, 32 (2007), 1208-1214.  doi: 10.1016/j.chaos.2005.11.071. [25] V. E. Zakharov, Collapse of Langmuir Waves, Soviet Physics-JETP, 35 (1972), 908-914. [26] Y. Zhang, H. H. Dong, X. E. Zhang and H. W. Yang, Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-like equation, Compu. Math. Appl., 73 (2017), 246-252.  doi: 10.1016/j.camwa.2016.11.009. [27] L. Zhang and C. M. Khalique, Classification and bifurcation of a class of second-order ODEs and its application to nonlinear PDEs, Discrete Contin. Dyn. Syst. Ser. S, 11 (2018), 759-772.  doi: 10.3934/dcdss.2018048. [28] L. Zhang, Y. Wang, C. M. Khlique and Y. Bai, Peakon and cuspon solutions of a generalized Camassa-Holm-Novikov equation, J. Appl. Anal. Compu., 8 (2018), 1938-1958. [29] J. Zhang, L. Zhang and Y. Bai, Stability and bifurcation analysis on a predator prey system with the weak allee effect, Mathematics, 7 (2019), 1-15.  doi: 10.3390/math7050432. [30] H. Zhao and W. Ma, Mixed lumpkink solutions to the KP equation, Compu. Math. Appl., 74 (2017), 1399-1405.  doi: 10.1016/j.camwa.2017.06.034. [31] Q. L. Zhao and X. Y. Li, A Bargmann system and the involutive solutions associated with a new 4-Order Lattice hierarchy, Anal. Math Phys., 6 (2016), 237-254.  doi: 10.1007/s13324-015-0116-2. [32] X. X. Zheng, Y. D. Shang and X. M. Peng, Orbital stability of solitary waves of the coupled Klein-Gordon-Zakharov equations, Math. Meth. Appl. Sci., 40 (2017), 2623-2633.  doi: 10.1002/mma.4187.
Phase portraits of system (13) with $b = 0$. (a) $d<0$ $\&$ $a>0$; (b) $d>0$ $\&$ $a<0$; (c) $d\leq0$ $\&$ $a<0$
Phase portraits of system (13) with $b>0$. (A) $d>0$, $a<0$ $\&$ $0<b<-\frac{4a^3}{27d^2}$; (B) $d<0$ $\&$ $a\geq0$ or $d\leq0$ $\&$ $a<0$
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