The Dynamics of Two Classes of Singular Nonlinear Traveling Wave Equations and Loop Solutions

Abstract

In this chapter, applying the qualitative theory of planar dynamical systems, we study the traveling solitary wave solutions and loop solutions of several nonlinear evolution equations. Dynamical behaviors and phase portraits are illustrated by numerical simulations in differential parametric regions. Three typical traveling wave equations are carefully investigated. All phase portraits of their corresponding traveling waves with different profiles are given. Analysis from a theoretical viewpoint of dynamical systems shows that the so-called loop- and inverted-loop-soliton solutions, as well as loop-periodic solutions reportedly existing in the above three equations, are merely visual illusion of numerical artifacts. To reveal the mystery, all the exact parametric representations of traveling wave solutions of the above three equations are derived in precise analytical forms.

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