Traveling waves and weak solutions for an equation with degenerate dispersion

We consider the following family of equations: ut = 2uuxxx − uxuxx + 2kuux. Here k 6= 0 is a constant and x ∈ [−L0, L0]. We demonstrate that for these equations: (a) there are compactly supported traveling wave solutions (which are in H) and (b) the Cauchy problem (with H initial data) possesses a weak solution which exists locally in time. These are the the first degenerate dispersive evolution PDE where (a) and (b) are known to hold simultaneously. Moreover, if k < 0 or L0 is not too large, the solution exists globally in time.