LAUSR

Abstract In this paper, the bounded traveling wave solutions of the modified water wave equations of which one dependent variable attains the singular value 2 c in finite or infinite time are investigated by using the bifurcation theory of planar dynamical systems. The line V = 2 c is the so-called singular line of the associated dynamical system and the results of this paper show that the solutions possess singularity if and only if their corresponding phase orbits intersect with this singular line. There are two types of solutions corresponding to these orbits intersecting with the singular line: smooth classical solutions and compact solutions possessing compact support in H l o c 1 ( R ) , which suggests that the existence of singular line breaks the uniqueness of solutions in H l o c 1 ( R ) space. There is a significant discovery from the investigation of the modified water wave equations that there are new type of solitary wave solutions approaching the singular value 2 c as time tends to infinite that correspond to some specific orbits connecting with singular lines of the associated traveling wave system, which refreshes and enriches the knowledge of the effects of singular lines on the traveling wave solutions to nonlinear wave equations. The explicit bounded smooth traveling wave solutions and compact solutions of the modified water wave equations are presented and simulated numerically.