A Kundu–nonlinear Schrödinger equation: Rogue waves, breathers, and mixed interaction solutions

Abstract

A Kundu–nonlinear Schrödinger equation that can be utilized to simulate the pulse propagation in optical fibers is researched in this paper. First, the Lax integrability of the above equation is proved and its modulational instability (i.e., the main mechanism for producing the rogue wave solutions and the breather solutions) is calculated. Subsequently, using the generalized perturbation (n,N−n)-fold Darboux transformation, the rogue waves, breathers, and mixed interaction solutions are acquired, as well as the impact of various parameters on the solutions is examined. In particular, when we assume that the coefficient of the equation is θ=mx+dt, some new wave structures are found based on parameter variations, such as the rotational separation of first-order rogue waves, scale-like structures generated by second-order breathers, etc., which offer novel ideas for producing different signals via optical fibers. Ultimately, the classification numbers of mixed solutions of rogue waves and breathers are provided, which can better observe how the two types of the local waves are combined, in accordance with the distributions of the increasing numbers of algebraic equations.