| Citation: |
Jing Jiang, Yixian Gao, Weipeng Zhang, Lan Yin. BIFURCATION OF TRAVELING WAVE SOLUTIONS OF THE $K(M, N)$ EQUATION WITH GENERALIZED EVOLUTION TERM[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 853-863. doi: 10.11948/2156-907X.20180035
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BIFURCATION OF TRAVELING WAVE SOLUTIONS OF THE $K(M, N)$ EQUATION WITH GENERALIZED EVOLUTION TERM
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Abstract
In this paper, by using bifurcation theory and methods of plane dynamic system, we investigate the bifurcations of the traveling wave system corresponding to the $K(m, n)$ equation with generalized evolution term. Under different parameter conditions, some exact explicit parametric representations of traveling wave solution are obtained. -
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References
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Jing Jiang, Yixian Gao, Weipeng Zhang, Lan Yin. BIFURCATION OF TRAVELING WAVE SOLUTIONS OF THE $K(M, N)$ EQUATION WITH GENERALIZED EVOLUTION TERM[J]. Journal of Applied Analysis & Computation, 2019, 9(3): 853-863. doi: 10.11948/2156-907X.20180035
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Figure 1.
$(g, c)$ -plane, where$L_{1}=L_1^+\cup L_1^-:g=0$ ,$L_{2}^{+}:c=\sqrt{-ag} ~(g < 0)$ . -
Figure 2. The phase portrait bifurcation of system (2.4) when
$a>0$ . -
Figure 3.
$(g, c)$ -plane, where$L_{1}=L_3^+\cup L_3^-:g=0$ ,$L_{4}^{-}:c=-\sqrt{-ag} ~(g < 0)$ . -
Figure 4. The phase portrait bifurcation of system (2.4) when
$a < 0$ . - Figure 5. The smooth periodic wave solution of the equation (3.2) with a=1, b=2, c=-4, g=1.
- Figure 6. The smooth periodic wave solution of the equation (3.3) when a=1, b=2, c=-4, g=1.