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Dynamics of Singular Traveling Wave Solutions of a Short Capillary-Gravity Wave Equation
Leta Temesgen Desta,Liu Wenjun,El Achab Abdelfattah
Keywords:Singular parabola, singular periodic wave solutions, multi-peaked periodic wave solutions, two-peaked solitary wave solutions.
      In this paper, dynamical behaviour of traveling wave solutions to a short capillary-gravity is analyzed by using the method of bifurcation. When the phase orbits intersects the singular parabola $y^2=\frac{2}{\lambda} \phi$ on the phase plane, then the trajectory creates a more 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 the 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.