A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations

Consider the mCH and mDP equations given by ut−uxxt+3u2ux=2uxuxx+uuxxx, and ut−uxxt+4u2ux=3uxuxx+uuxxx, respectively. A new characteristic of solitary wave solutions, bell-shaped solitary wave and peakon coexist for the same wave speed, is found. Previous some results are extended. A conjecture is clarified.