High-order rogue waves of a long wave-short wave model

TL;DR

This paper derives exact higher-order rogue wave solutions for a long wave-short wave interaction model, classifies their patterns, and links their existence to modulation instability conditions.

Contribution

It introduces a determinant-based method to obtain higher-order rogue waves and classifies their patterns within the long wave-short wave model.

Findings

Fundamental short wave rogue waves are classified into three patterns: bright, intermediate, dark.

Long wave rogue waves are always bright.

Higher-order rogue waves are superpositions of fundamental rogue waves.

Abstract

The long wave-short wave model describes the interaction between the long wave and the short wave. Exact higher-order rational solution expressed by determinants is calculated via the Hirota's bilinear method and the KP hierarchy reduction. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate and dark ones, whereas the rogue wave for the long wave is always bright type. The higher-order rogue waves correspond to the superposition of fundamental rogue waves. The modulation instability analysis show that the condition of the baseband modulation instability where an unstable continuous-wave background corresponds to perturbations with infinitesimally small frequencies, coincides with the condition for the existence of rogue-wave solutions.