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

Article Contents

2020, Volume 13, Issue 10: 2927-2939. Doi: 10.3934/dcdss.2020214

This issue Previous Article Smooth and singular traveling wave solutions for the Serre-Green-Naghdi equations Next Article A study on lump solutions to a (2+1)-dimensional completely generalized Hirota-Satsuma-Ito equation

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
^* Corresponding author: Lijun Zhang

Received: November 1, 2018

Revised: July 1, 2019

Early access: December 21, 2019

Published online: December 21, 2019

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

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Abstract

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.

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

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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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