The interaction solitons for the complex short pulse equation

We study the complex short pulse equation in optical fibers. By a method associated with the Darboux matrix operator, the N-fold Darboux transformation on the matrix and scalar solutions for Lax pair of the complex short pulse equation are given in terms of the quasideterminant and determinant, respectively. As applications of the Darboux transformation, multiply smooth soliton, loop soliton, cuspon soliton, breather soliton and rogue wave solutions of the complex short pulse equation are given explicitly by different seed solutions. Further, one-, two-, and three-soliton solutions are investigated in details, and shown to interesting phenomenon of energy redistribution in different soliton interactions, which shows smooth soliton, cuspon soliton, and loop soliton are the fastest runner, the middling runner, and the slowest runner, respectively, among the wave propagation. In particular, the breather soliton and rogue wave solutions are two kinds of novel solutions.