Multi-soliton, multi-breather and higher-order rogue wave solutions to the complex short pulse equation

In the present paper, we are concerned with the general localized solutions for the complex short pulse equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the $N$-bright soliton solution in a compact determinant form, then the $N$-breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first- and second-order rogue wave solutions are given explicitly and illustrated by graphs. The asymptotic analysis is performed rigorously for both the $N$-soliton and the $N$-breather solutions. All three forms of the localized solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation is different from the one to the nonlinear Schr\"odinger (NLS) equation, which could be a cusponed- or a looped one.