Singular solitary waves associated with homoclinic orbits

Different from what we have expected before, when a homoclinic orbit intersects with a quadratic singular curve on the topological phase plane derived from a generalized KdV equation, corresponding to the homoclinic orbit, there exist a few types of solitary waves, including peakons and antipeakons as well as periodic waves and furthermore, new types of solitary waves with peaks. The investigation shows that, when a trajectory along the homoclinic orbit moves at the intersection points between the homoclinic orbit and the singular curve, it may jump between these intersection points, which forms peaks on the waves, implying the nonsmooth solitary waves occur.