Existence of exotic waves for the nonlinear dispersive mKdV equation

The nonlinear dispersive mKdV equation ut+(u3)x+(u2)xxx=0 is proved to be Painleve integrable. Detailed classification of its travelling waves under certain parameter conditions are obtained by the improved qualitative analysis method. Abundant type of solutions are shown to exist including exotic traveling waves, peaked waves, compacted waves, looped and cusped waves, very particular composite waves having two singular points.