Solitons, breathers and rogue waves of the Yajima–Oikawa-Newell long wave–short wave system

Abstract

In this paper, we consider the recently-introduced Yajima–Oikawa–Newell (YON) system describing the nonlinear resonant interaction between a long wave and a short wave. It extends and generalises the Yajima–Oikawa (YO) and the Newell (N) systems, which can be obtained from the YON system for special choices of the two non-rescalable, arbitrary parameters that it features. Remarkably, for any choice of these latter constants, the YON system is integrable, in the sense of possessing a Lax pair. New families of solutions, including the bright and dark multi-solitons, as well as the breathers and the higher-order rogue waves are systematically derived by means of the $\tau$-function reduction technique for the two-component KP and the KP-Toda hierarchies. In particular, we show that the condition that the wave parameters have to satisfy for the rogue wave solution to exist coincides with the prediction based on the stability spectra for base-band instability of the plane wave solutions. Several examples from each family of solutions are given in closed form, along with a discussion of their main properties and behaviours.