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Bifurcations and New Traveling Wave Solutions for the Nonlinear Dispersion Drinfel'd-Sokolov ($D(m,n)$) System
  
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KeyWord:$D(m,n)$ system; Solitary wave solution; Periodic wave solution; Compacton; Peakon
Author NameAffiliation
Ronghua Cheng School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, Jiangsu 210044, China
School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China 
Zhaofu Luo School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China 
Xiaochun Hong School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China 
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Abstract:
      In this paper, we employ the theory of the planar dynamical system to investigate the dynamical behavior and bifurcations of solutions of the traveling systems of the $D(m,n)$ equation. On the basis of the previous work of the reference \cite{zhang}, we obtain the solitary cusp waves solutions (peakons and valleyons), breaking wave solutions (compactons) and other periodic cusp wave solutions. Morever, we make a summary of exact traveling wave solutions to the $D(m,n)$ system including all the solutions which have been found from the references \cite{Deng,Xie,zhang}.