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₏, ₑ^s with 1 ≤ p, r ≤ ∞, s > max 1 + 1 / p, 3 / 2 s >max\1+1/p, 3/2\ or B 2, 1 3 / 2 B₂, ₁^3/2, and then prove the ill-posedness of the solution in B 2, ∞ 3 / 2 B₂, ^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.