Akhmediev Breathers and Kuznetsov–Ma Solitons in the Cubic-Quintic Nonlinear Schrödinger Equation

Finite background solitons are a significant area of research in nonlinear dynamics, as they are commonly found in various complex physical systems. Understanding how finite background solitons are generated and determining the conditions required for their excitation is crucial for detecting and applying dynamic characteristics. We used the Darboux transformation method to obtain explicit analytical solutions for the Akhmediev breather, the Kuznetsov-Ma soliton, and the Peregrine soliton of the cubic-quintic nonlinear Schrödinger equation. This equation is typically used as a model to control the propagation of ultrashort pulses in highly nonlinear optical fibers. We also provide the conditions required for the existence of these different breather solutions and discuss their interesting dynamical properties, such as oscillation period, propagation direction, and peak amplitude. We systematically discuss the excitation conditions and phase diagrams of the breathers by analyzing modulation instability. These results and associated formulas can also be extended to vector or multi-component systems, the breathing dynamics of which remain to be explored.