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Solitary waves are coincided with separatrices, which surround an
equilibrium point with characteristics like a center, a sink, or a source. The
existence of closed orbits in phase plane predicts the existence of such an
equilibrium point. If there exists another saddle point near that equilibrium
point, separatrix orbit appears. In order to prove the existence of solution for
any kind of boundary value problem, we need to apply the fixed-point
theorems. The Schauder’s fixed-point theorem was used to show that there
exists at least one nontrivial solution for equation of wave motion in arteries.
The equation of wave motion in arteries has a nonlinear character, and the
amplitude of the wave depends on the wave velocity. There is no general
analytical or straightforward method for prediction of the amplitude of
solitary waves. Therefore, the solution must be found by numerical or non-straightforward
methods. The methods of saddle point trajectory, escape-time,
and escape-energy are introduced and shown that they are applicable
methods with enough accuracy. Application of any of these approximate
methods depends on the equation of motion, and the user preference.
Applying a phase plane analysis, it was shown that the domain of periodic
solution is surrounded by a separatrix. The separatrix is coincident with the
desired solitary wave. The amplitude of the solitary wave is the most
important characteristic of the wave, and will be predicted with each of the
above methods.