# 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  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 & Continuous Dynamical Systems - S, 2020, 13 (10) : 2927-2939. doi: 10.3934/dcdss.2020214
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##### References:
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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