| Citation: |
Nail Turhan, Ali Deliceoğlu, Yusuf Pandır. BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED KLEIN- GORDON EQUATION WITH BETA FRACTIONAL DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 212-254. doi: 10.11948/20260014
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BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED KLEIN- GORDON EQUATION WITH BETA FRACTIONAL DERIVATIVES
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Abstract
In this paper, we consider the exact traveling wave solutions of the generalized Klein-Gordon equation with beta fractional derivatives. By using the bifurcation theory of dynamical systems, we derive the bifurcation behavior of its phase portraits under various parameter settings. For some special level curves, we obtain all possible traveling wave solutions such as periodic wave solutions, dark and bright solitary wave solutions, kink and anti-kink solutions, and some unbounded wave solutions. The impacts of beta fractional orders are analyzed using graphs, with a focus on their impact on temporal delay, broadening, and the oscillation period. Furthermore, the present work generalizes and extends earlier findings in the literature. These results contribute to a deeper understanding of the dynamics of the generalized Klein-Gordon equation model with beta fractional derivatives.
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References
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Nail Turhan, Ali Deliceoğlu, Yusuf Pandır. BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED KLEIN- GORDON EQUATION WITH BETA FRACTIONAL DERIVATIVES[J]. Journal of Applied Analysis & Computation, 2027, 17(1): 212-254. doi: 10.11948/20260014
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Figure 1.
The phase portraits of system (3.4) for
$ c>0 $ $ p>0 $ $ n<0 $ -
Figure 2.
The phase portraits of system (3.4) for
$ c <0 $ $ p>0 $ $ n<0 $ -
Figure 3.
The phase portraits of system (3.4) for
$ b=0 $ $ p>0 $ $ n>0 $ -
Figure 4.
The phase portraits of system (3.4) for
$ c=0 $ $ p>0 $ $ n<0 $ -
Figure 5.
The level curves defined by Eq. (3.6) for
$ h = 0 $ $ p > 0 $ $ m = \frac{\mu n}{2} $ -
Figure 6.
The variations of the level curves defined by
$ H(v, y)=h $ $ \Delta > 0 $ $ p > 0 $ $ n < 0 $ -
Figure 7.
The phase portraits of system (4.5) for
$ p>0 $ $ \Delta_1>0 $ $ mn>0 $ -
Figure 8.
The phase portraits of system (4.5) for
$ p>0 $ -
Figure 9.
The level curves defined by Eq. (4.6) for
$ h=0 $ $ p>0 $ $ n=\frac{m(\mu+1)}{\mu} $ -
Figure 10.
The level curves defined by Eqs. (4.13) and (4.31) for
$ h\ne0 $ -
Figure 11.
Graphical representations of the periodic wave solution with singularities given in Eq. (5.1) with parameters
$ \alpha =a=c=k=1 $ $ b=6 $ $ \lambda=2 $ $ n=2 $ $ m=3 $ $ \xi_0=0 $ $ \beta_2 = 1 $ $ 3D $ $ \beta_1 = 1 $ $ 0.8 $ $ 0.65 $ $ 2D $ $ \beta_1 $ -
Figure 12.
Graphical representations of the periodic wave solution with singularities given in Eq. (5.1) with parameters
$ \alpha =a=c=k=1 $ $ b=6 $ $ \lambda=2 $ $ n=2 $ $ m=3 $ $ \xi_0=0 $ $ \beta_1 = 1 $ $ 3D $ $ \beta_2 = 1 $ $ 0.8 $ $ 0.65 $ $ 2D $ $ \beta_2 $ -
Figure 13.
Graphical representations of the solitary wave solution given in (5.2) with parameters
$ \alpha = c = k = 1 $ $ a = -3 $ $ b = -2 $ $ \lambda = 2 $ $ n = -2 $ $ m = 2 $ $ \xi_0=0 $ $ \beta_2 = 1 $ $ 3D $ $ \beta_1 = 1 $ $ 0.8 $ $ 0.6 $ $ 2D $ $ \beta_1 $ -
Figure 14.
Graphical representations of the periodic wave solution given in (5.3) with parameters
$ \alpha =m=c=k=1 $ $ a=2 $ $ b=3 $ $ \lambda=2 $ $ n=3 $ $ \xi_0=0 $ $ \beta_2 = 1 $ $ 3D $ $ \beta_1 = 0.95 $ $ 0.85 $ $ 0.7 $ $ 2D $ $ \beta_1 $ -
Figure 15.
Graphical representations of the dark soliton solution given in (5.4) with parameters
$ \alpha =m=k=1 $ $ a=-2 $ $ b=-3 $ $ c=-1 $ $ \lambda=2 $ $ n=3 $ $ \xi_0=0 $ $ \beta_2 = 1 $ $ 3D $ $ \beta_1 = 0.95 $ $ 0.8 $ $ 0.65 $ $ 2D $ $ \beta_1 $ -
Figure 16.
Graphical representations of the kink wave solution given in (5.5) with parameters
$ \alpha=m=k=1 $ $ a=-3 $ $ b=-4 $ $ c=-1 $ $ \lambda=2 $ $ n=3 $ $ \xi_0=0 $ $ \beta_2 = 1 $ $ 3D $ $ \beta_1 = 0.9 $ $ 0.75 $ $ 0.6 $ $ 2D $ $ \beta_1 $