Two kinds of peaked solitary waves of the KdV, BBM and Boussinesq equations

It is well-known that the celebrated Camassa-Holm equation has the peaked solitary waves, which have been not reported for other mainstream models of shallow water waves. In this letter, the closed-form solutions of peaked solitary waves of the KdV equation, the BBM equation and the Boussinesq equation are given for the first time. All of them have either a peakon or an anti-peakon. Each of them exactly satisfies the corresponding Rankine-Hogoniot jump condition and should be understood as weak solution. Therefore, the peaked solitary waves might be common for most of shallow water wave models, no matter whether or not they are integrable and/or admit breaking-wave solutions.