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

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>r</m:mi> </m:mrow> <m:mrow> <m:mi>s</m:mi> </m:mrow> </m:msubsup> </m:math> Bp,rˢ with 1 ≤ p , r ≤ ∞, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>s</m:mi> <m:mo>></m:mo> <m:mi>max</m:mi> <m:mfenced close="}" open="{"> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>/</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mn>3</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mfenced> </m:math> s >max\1+1/p,3/2\ or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mn>2,1</m:mn> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:math> B2,13/2 , and then prove the ill-posedness of the solution in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>B</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi>∞</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> </m:math> 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.