| Citation: |
Temesgen Desta Leta, Wenjun Liu, Abdelfattah El Achab. DYNAMICS OF SINGULAR TRAVELING WAVE SOLUTIONS OF A SHORT CAPILLARY-GRAVITY WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1191-1207. doi: 10.11948/20200032
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DYNAMICS OF SINGULAR TRAVELING WAVE SOLUTIONS OF A SHORT CAPILLARY-GRAVITY WAVE EQUATION
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Abstract
In this paper, dynamical behaviour of traveling wave solutions to a short capillary-gravity equation is analyzed by using the method of bifurcation. When the phase orbits intersects the singular parabola $ y^2 = 2\phi/\lambda $ on the phase plane, then the trajectories create a weaker wave fronts than the regular traveling wave solutions. By using proper Euler transformations, we reformulate the model as a singular chaotic problem, which can then be analyzed using the singularity study. We prove existence of three types of physically realistic traveling wave solutions to the case of small diffusion for the first time, two-peaked solitary waves, three-peaked and multi-peaked periodic wave solutions.
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References
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Temesgen Desta Leta, Wenjun Liu, Abdelfattah El Achab. DYNAMICS OF SINGULAR TRAVELING WAVE SOLUTIONS OF A SHORT CAPILLARY-GRAVITY WAVE EQUATION[J]. Journal of Applied Analysis & Computation, 2021, 11(3): 1191-1207. doi: 10.11948/20200032
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Figure 1. Bifurcations of phase portraits of system (2.2) when
$ \rho>0 $ and$ \lambda<0. $ -
Figure 2. Bifurcations of phase portraits of system (2.2) when
$ \rho<0 $ and$ \lambda>0. $ -
Figure 3. Bifurcations of phase portraits of system (2.2) when
$ \rho<0 $ and$ \lambda\geq0. $ -
Figure 4. Bifurcations of phase portraits of system (2.2) when
$ \rho\rightarrow0 $ and$ \lambda = 0. $ -
Figure 5. Bifurcations of phase portraits of system (8) when
$ \rho>0 $ and$ \lambda>0. $ -
Figure 6. As
$ h\rightarrow0, $ a smooth periodic wave evolved into singular periodic wave. - Figure 7. Wave profile corresponding to exact wave solution of Eq. (3.10).
- Figure 8. Wave profiles of system (2.2)
- Figure 9. Traveling wave solution of system (2.5). 9(a) A breaking wave of Eq. (3.43); 9(b) A peakon wave solution and 9(c) An anti-peakon wave solution of Eq. (3.44)
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Figure 10. As
$ h $ goes from$ h_0 $ $ \rightarrow $ $ h_s, $ the smooth solitary wave evolves into a singular periodic wave.