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

Lijun Zhang; Peiying Yuan; Jingli Fu; Chaudry Masood Khalique

doi:10.3934/dcdss.2020214

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

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

Lijun Zhang ^1,2,, , Peiying Yuan ^3, , Jingli Fu ^4, and Chaudry Masood Khalique ^5,

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.

Keywords: Bifurcation, traveling wave solutions, the Zakharov-Rubenchik equation, dynamical system method.

Mathematics Subject Classification: 35C07, 34G20, 34C23.

Citation: Lijun Zhang, Peiying Yuan, Jingli Fu, Chaudry Masood Khalique. Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (10) : 2927-2939. doi: 10.3934/dcdss.2020214

References:

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[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. Google Scholar
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[13]	J. Li, Singular Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. Google Scholar
[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. Google Scholar
[15]	F. Linares and C. Matheus, Well posedness for the 1D Zakharov-Rubenchik system, Adv. Differential Equations, 14 (2009), 261-288. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	V. E. Zakharov, Collapse of Langmuir Waves, Soviet Physics-JETP, 35 (1972), 908-914. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[2]	P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, Springer-Verlag, New York-Berlin, 1971. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[13]	J. Li, Singular Traveling Wave Equations: Bifurcations and Exact Solutions, Science Press, Beijing, 2013. Google Scholar
[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. Google Scholar
[15]	F. Linares and C. Matheus, Well posedness for the 1D Zakharov-Rubenchik system, Adv. Differential Equations, 14 (2009), 261-288. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[25]	V. E. Zakharov, Collapse of Langmuir Waves, Soviet Physics-JETP, 35 (1972), 908-914. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

Figure 1. Phase portraits of system (13) with $ b = 0 $. (a) $ d<0 $

$ \& $

$ a>0 $; (b) $ d>0 $

$ \& $

$ a<0 $; (c) $ d\leq0 $

$ \& $

$ a<0 $

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Figure 2. 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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