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

•We construct a generalized Darboux transformation (gDT) to the CSP and CCD equations.•We derive multi-bright soliton solution to the CSP equation based on the gDT.•We construct a single and multi-breather solutions to the CSP equation.•First and higher order rogue wave solutions to the CSP equation are constructed.

In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the NN-bright soliton solution in a compact determinant form, the NN-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 analyzed. The asymptotic analysis is performed rigorously for both the NN-soliton and the NN-breather solutions. All three forms of the analytical 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 can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.