A matrix Yajima–Oikawa long-wave-short-wave resonance equation, Darboux transformations and rogue wave solutions

Abstract

Based on an introduced (2m+n)×(2m+n) matrix spectral problem, a matrix Yajima–Oikawa long-wave-short-wave resonance equation is proposed, which can be reduced to an (n+1)-component Yajima–Oikawa long-wave-short-wave resonance equation. Multi-fold generalized Darboux transformations for these two equations are constructed by using the gauge transformation between Lax pairs and their Riccati equations. Every solution to the Riccati equations can be transformed into a new solution of the matrix Yajima–Oikawa long-wave-short-wave resonance equation through the Darboux transformations. As an application of the obtained Darboux transformation, first, we derive high-order rogue wave solutions of the Yajima–Oikawa long-wave-short-wave resonance equation. Second, we obtain explicit solutions for the lower-order matrix Yajima–Oikawa long-wave-short-wave resonance equation, including soliton solutions, rogue wave solutions, wave solutions that are constant along the x-direction, and wave solutions traveling at varying speeds.