Bifurcations of traveling wave solutions for an integrable equation

This paper deals with the following equation m(t)=(1/2)(1/m(k))(xxx)-(1/2)(1/m(k))(x), which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-1/2, 1/2,2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions. (C) 2010 American Institute of Physics. [doi: 10.1063/1.3385777]