The dynamics of the breather, degenerate solutions and dark peakon solutions for the focusing and defocusing complex short pulse equations

Abstract This paper proposes a uniform N-fold Darboux transformation (DT) for both the focusing and defocusing complex short pulse (CSP) equations, which is expressed in determinant and compact form. Through the application of uniform DT, we systematically constructed the breather under vanishing boundary condition (VBC) in the focusing case and classified its dynamics. Additionally, the degenerate uniformDT and degenerate breather (i.e., breather-positon) under VBC are also obtained by performing Taylor asymptotic expansion. The generation of breather under VBC is related to the "gravitation-repulsion effect", "partial annihilation effect" and "resonance effect" between solitons. We further analyzed the interaction between breather, soliton and degenerate solutions, and proved that degenerate solutions are transparent when colliding with soliton and breather. Consequently, degenerate solutions are also referred to as super-reflectionless potentials. For the defocusing case, we derive multi-dark smooth soliton and dark peakon solutions under the non-vanishing boundary condition (NVBC) from the compact form of the uniform DT, utilizing a specific limit. Furthermore, a single dark soliton’s dynamic classification and asymptotic behavior were studied using the zeros analysis method and asymptotic analysis respectively. Finally, by investigating the interactions between twoand three-dark soliton solutions, we find that dark solitons exhibit greater stability.