| Citation: |
Zhenjie Niu, Zenggui Wang. BIFURCATION AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR DISPERSIVE MK(M, N) EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2866-2875. doi: 10.11948/20210023
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BIFURCATION AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR DISPERSIVE MK(M, N) EQUATION
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Abstract
This paper investigated the generalized nonlinear dispersive mK(m, n) equation by the planar dynamical systems method, the bifurcations of the system with different parameter region of this equation are presented. Moreover, we find different kinds of exact explicit solutions like peak type solutions, periodic wave solutions and valley type solutions.
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References
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Zhenjie Niu, Zenggui Wang. BIFURCATION AND EXACT TRAVELING WAVE SOLUTIONS FOR THE GENERALIZED NONLINEAR DISPERSIVE MK(M, N) EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2866-2875. doi: 10.11948/20210023
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Figure 1. The bifurcation set of system(1.2).(a)
$ m-n = 2k+1 $ is odd; (b)$ m-n = 2k+2 $ is even -
Figure 2. The phase portraits of system (2.5) with
$ m-n = 2k+1 $ . (a)$ a>0, c>0 $ ; (b)$ a<0, c>0 $ ; (c)$ a>0, c<0 $ ; (d)$ a<0, c<0 $ - Figure 3. The phase portraits of system (2.5) with m − n = 2k. (a) a > 0, c > 0;(b) a < 0, c > 0; (c) a > 0, c < 0; (d) a < 0, c < 0
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Figure 4. The phase portraits of system (2.5) with
$ m-n = 2k+1 $ . (a)$ (c, g)\in A(1) $ ; (b)$ (c, g)\in A(2) $ ; (c)$ (c, g)\in A(3) $ ; (d)$ (c, g)\in A(4) $ ; (e)$ (c, g)\in A(5) $ ; (f)$ (c, g)\in A(6) $ -
Figure 5. The phase portraits of system (2.5) with
$ m-n = 2k+2 $ . (a)$ (c, g)\in $ B(1); (b)$ (c, g)\in $ B(2); (c)$ (c, g)\in $ B(3) -
Figure 6. (a) solution(3.1) when
$ a>0 $ ; (b) solution (3.1) when$ a<0 $ ; (c) solution (3.2) when$ a>0 $ ; (d) solution (3.2) when$ a<0 $ ; (e) solution (3.3) when$ u(\xi)>0 $ ; (f) solution (3.3) when$ u(\xi)<0 $