Higher-order rogue wave solutions to the Kadomtsev–Petviashvili 1 equation

Highlights: • A new family of eigenfunctions associated with the same seeding solution. • The Higher-order RW solutions expressed by exponential–rational functions. • The heights of RWs increase with increasing $n$, but the formation speeds decrease. We construct a family of eigenfunction solutions of the Lax pair of the Kadomtsev–Petviashvili 1 (KP1) equation, which is expressed in terms of the Taylor coefficients of a fundamental exponential function associated with the Lax pair. Using the binary Darboux transformation, the higher-order rogue waves on a solitonic background of the KP1 equation are obtained by multiple soliton interactions. In fact, these solitons can evolve to significant and strongly localized transient waves during directional evolution, similarly to KP2 shallow-water interactions. We conjecture that the $n\text{th}$-order rogue wave solution evolves in the form of a triangular extreme wave pattern that consists of $\frac{n(n+1)}{2}$ solitonic lumps in the intermediate time, while only $n+1$ parallel line solitons possessing equal height are present before and after the initiation of collision dynamics. Such fascinating higher-order rogue waves have not yet been reported in the context of this $2+1$ dimensional integrable system. These exact solutions are particularly relevant for the fundamental understanding of extreme wave events in a variety of physical systems, such as plasma and solids.