On the solarity waves in arteries

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 nonstraightforward methods. The methods of saddle point trajectory, escapetime, 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.