| Citation: |
Jinsen Zhuang, Yan Zhou. BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE EQUIVALENT COMPLEX SHORT-PULSE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 795-815. doi: 10.11948/20190408
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BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE EQUIVALENT COMPLEX SHORT-PULSE EQUATIONS
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Abstract
In this paper, we study the traveling wave solutions for a complex short-pulse equation of both focusing and defocusing types, which governs the propagation of ultrashort pulses in nonlinear optical fibers. It can be viewed as an analog of the nonlinear Schrödinger (NLS) equation in the ultrashort-pulse regime. The corresponding traveling wave systems of the equivalent complex short-pulse equations are two singular planar dynamical systems with four singular straight lines. By using the method of dynamical systems, bifurcation diagrams and explicit exact parametric representations of the solutions are given, including solitary wave solution, periodic wave solution, peakon solution, periodic peakon solution and compacton solution under different parameter conditions. -
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References
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Jinsen Zhuang, Yan Zhou. BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE EQUIVALENT COMPLEX SHORT-PULSE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(2): 795-815. doi: 10.11948/20190408
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Figure 1. Bifurcations of phase portraits of system (1.9) in the
$ (\psi, y) $ -phase plane for a fixed$ \beta>0 $ -
Figure 2. Some level curves of system (1.9) defined by
$ H_+(\psi, y) = h $ -
Figure 3. Two solitary wave solutions of system (1.9) given by
$ H_+(\psi, y) = h_3 $ when$ \delta = -1.12 $ -
Figure 4. Solitary wave, peakon and periodic peakon of system (1.9) given by
$ H_+(\psi, y) = h_s $ -
Figure 5. Some level curves of system (1.9) defined by
$ H_+(\psi, y) = h_s $ -
Figure 6. Periodic peakon solutions of system (1.9) given by
$ H_+(\psi, y) = h_s $ -
Figure 7. Periodic solutions of system (1.9) given by
$ H_+(\psi, y) = h_s $ -
Figure 8. Some level curves of system (1.9) defined by
$ H_+(\psi, y) = h $ -
Figure 9. Compacton solution families of system (1.9) defined by
$ H_+(\psi, y) = h $ -
Figure 10. Bifurcations of phase portraits of system (1.10) in the
$ (\psi, y) $ -phase plane