| Abstract: |
| In this paper, the bounded traveling wave solutions of the modified water wave
equations of which one dependent variable attains the singular value 2c in finite or
infinite time are investigated by using the bifurcation theory of planar dynamical
systems. The line V = 2c 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
H1loc(R), which suggests that the existence of singular line breaks the uniqueness of
solutions in H1loc(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 2c 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. |
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