School of Mathematical and Statistical Sciences Faculty Publications and Presentations

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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 {\sf 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 Bsp,r with 1≤p,r≤∞ , s>max{1+1/p,3/2} or B3/22,1 , and then prove the ill-posedness of the solution in B3/22,∞ . Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if m0(x)=u0−u0xx≥(≢)0 , then the corresponding solution exists globally, meanwhile, if m0(x)≤(≢)0 , then the corresponding solution blows up in a finite time.

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