Darboux transformations for a matrix long‐wave–short‐wave equation and higher‐order rational rogue‐wave solutions

Abstract

A new matrix long‐wave–short‐wave equation is proposed with the of help of the zero‐curvature equation. Based on the gauge transformation between Lax pairs, both onefold and multifold classical Darboux transformations are constructed for the matrix long‐wave–short‐wave equation. Resorting to the classical Darboux transformation, a multifold generalized Darboux transformation of the matrix long‐wave–short‐wave equation is derived by utilizing the limit technique, from which rogue wave solutions, in particular, can be obtained by employing the generalized Darboux transformation. As applications, we obtain rogue‐wave solutions of the long‐wave–short‐wave equation and some explicit solutions of the three‐component long‐wave–short‐wave model, including soliton solutions, breather solutions, the first‐order and higher‐order rogue‐wave solutions, and others by using the generalized Darboux transformation.