| Citation: |
Jibin Li, Maoan Han. EXACT PEAKON SOLUTIONS GIVEN BY THE GENERALIZED HYPERBOLIC FUNCTIONS FOR SOME NONLINEAR WAVE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1708-1719. doi: 10.11948/20200139
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EXACT PEAKON SOLUTIONS GIVEN BY THE GENERALIZED HYPERBOLIC FUNCTIONS FOR SOME NONLINEAR WAVE EQUATIONS
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Abstract
In 1993, Camassa and Holm drived a shallow water equation and found that this equation has a peakon solution with the form $ \phi(\xi) = ce^{-|\xi|} $. In this paper, we show that three nonlinear wave systems have peakon solutions which needs to be represented as generalized hyperbolic functions. For the existence of these solutions, some constraint parameter conditions are derived. -
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References
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Jibin Li, Maoan Han. EXACT PEAKON SOLUTIONS GIVEN BY THE GENERALIZED HYPERBOLIC FUNCTIONS FOR SOME NONLINEAR WAVE EQUATIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(4): 1708-1719. doi: 10.11948/20200139
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Figure 1. The bifurcations of phase portraits of system (1.2) when
$ g = 0 $ . -
Figure 2. The curve triangle defined by
$ H_1(\phi,y) = h_s $ of system (2.2). -
Figure 3. The curve triangle defined by
$ H_2(\phi,y) = h_s $ of system (3.3). (a) Phase portrait when$ \frac{-q+\sqrt{\Delta}}{2p} <a<\frac{-q-\sqrt{\Delta}}{2p} $ . (b) Level curves defined by$ H_2(\phi,y) = 0 = h_s $ . (c) Phase portrait when$ a>\frac{-q+\sqrt{\Delta}}{2p} $ . (d) Level curves defined by$ H_2(\phi,y) = h_2 = h_s $ . -
Figure 4. Phase portraits and curve triangle defined by
$ H_3(\phi, y) = h_s $ of system (4.1). (a)$ h_1<h_2<h_s = h_3<h_4, \phi_4<\frac{c+\mu}{\sigma}, \sigma<1. $ (b)$ h_1<h_2<h_3<h_4 = h_s, c<\frac{c+\mu}{\sigma}<\phi_3, \sigma<1. $ (c) In Fig. 4 (a), the level curves defined by$ H_3(\phi, y) = h_s $ . (d) Peakon solution of system (4.1) corresponding to the curve triangle in Fig. 4 (c).