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.

Multisoliton and general Wronskian solutions of a variable‐coefficient Korteweg–de Vries equation

“…The construction of explicit solutions is an important part of the investigation of these equations. There are several effective methods to find exact solutions for these equations, such as inverse scattering transformation [1][2][3], Darboux transformation [4][5][6][7][8][9], Hirota bilinear method, and Wronskian technique [10][11][12][13][14][15][16]. Among them, Hirota bilinear method is a direct approach to construct exact solutions of nonlinear evolution equations.…”

Time-fractional Davey–Stewartson equation: Lie point symmetries, similarity reductions, conservation laws and traveling wave solutions

“…In the last few decades, researchers have paid much attention to the study of NPDEs, including their dynamic properties and solutions. A number of methods were put forward to acquire the solutions of NPDEs, for example, the homogeneous balance method [10][11][12][13], inverse scattering transform [14][15][16][17][18], Darboux transformation [19][20][21][22][23][24], Hirota's bilinear method [25][26][27][28][29][30][31][32][33], Riemann-Hilbert approach [34][35][36][37][38][39][40][41][42][43], nonlocal symmetry method and so on [44][45][46][47][48][49]. However, for most NPDEs, the solutions are difficult to obtain due to their complex expressions.…”