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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Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations
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 |
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 and Continuous Dynamical Systems - S,
2020, 13
(10)
: 2927-2939.
doi: 10.3934/dcdss.2020214
![]()
References:
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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.
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Exp-function method for nonlinear wave equations, Chaos Solitons Fractals, 30 (2006), 700-708.
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New interaction solutions for the (3+1)-dimensional Jimbo-Miwa equation, Compu. Math. Appl., 76 (2018), 2141-2147.
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Exact and numerical solitary wave solutions of generalized Zakharov equation by the variational iteration method, Chaos Soliton Fractals, 36 (2008), 309-313.
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General energy decay for a degenerate viscoelastic Petrovsky-type plate equation with boundary feedback, J. Appl. Anal. Compu., 8 (2018), 390-401.
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A new integrable symplectic map by the binary nonlinearization to the super AKNS system, J. Geom. Phys., 121 (2017), 123-137.
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F. Li and Q. Gao,
Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Compu., 274 (2016), 383-392.
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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.
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Solutions of a discrete integrable hierarchy by straightening out of its continuous and discrete constrained flows, Anal. Math. Phys., 9 (2019), 465-481.
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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.
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F. Oliveira,
Stability of the solitons for the one-dimensional Zakharov-Rubenchik equation, Phys. D, 175 (2003), 220-240.
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F. Oliveira,
Adiabatic limit of the Zakharov-Rubenchik Equation, Reports Math. Phys., 61 (2008), 13-27.
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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.
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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. |
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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