Rogue peakon, posedness and blow-up phenomenon for an integrable Camassa–Holm type equation

Abstract In this paper, we study an integrable Camassa–Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution – called rogue peakon , that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space B p , r s ${B}_{p,r}^{s}$ with 1 ≤ p , r ≤ ∞, s > max 1 + 1 / p , 3 / 2 $s{ >}\mathrm{max}\left\{1+1/p,3/2\right\}$ or B 2,1 3 / 2 ${B}_{2,1}^{3/2}$ , and then prove the ill-posedness of the solution in B 2 , ∞ 3 / 2 ${B}_{2,\infty }^{3/2}$ . Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if m 0 ( x ) = u 0 − u 0xx ≥ (≢)0, then the corresponding solution exists globally, meanwhile, if m 0 ( x ) ≤ (≢)0, then the corresponding solution blows up in a finite time.