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

2019 ◽  
Vol 43 (2) ◽  
pp. 948-967
Author(s):  
Ruomeng Li ◽  
Xianguo Geng ◽  
Bo Xue
Keyword(s):  
Wave Equation ◽  
Rogue Wave ◽  
Short Wave ◽  
Higher Order ◽  
Darboux Transformations ◽  
Wave Solutions ◽  
Long Wave ◽  
Rogue Wave Solutions

scholarly journals On a Vector Modified Yajima–Oikawa Long-Wave–Short-Wave Equation

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 958
Author(s):  
Xianguo Geng ◽  
Ruomeng Li
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Rogue Wave ◽  
Riccati Equations ◽  
Short Wave ◽  
Lax Pair ◽  
Darboux Transformations ◽  
Wave Solutions ◽  
Zero Curvature ◽  
Long Wave

A vector modified Yajima–Oikawa long-wave–short-wave equation is proposed using the zero-curvature presentation. On the basis of the Riccati equations associated with the Lax pair, a method is developed to construct multi-fold classical and generalized Darboux transformations for the vector modified Yajima–Oikawa long-wave–short-wave equation. As applications of the multi-fold classical Darboux transformations and generalized Darboux transformations, various exact solutions for the vector modified long-wave–short-wave equation are obtained, including soliton, breather, and rogue wave solutions.

Breather and rogue wave solutions of an extended nonlinear Schrödinger equation with higher-order odd and even terms

2018 ◽  
Vol 32 (26) ◽  
pp. 1850309 ◽  
Author(s):  
Dan Su ◽  
Xuelin Yong ◽  
Yanjiao Tian ◽  
Jing Tian
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Rogue Wave ◽  
Higher Order ◽  
Nonlinear Schrodinger Equation ◽  
Spectral Parameters ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Rogue Wave Solutions

In this paper, an extended nonlinear Schrödinger equation with higher-order odd (third-order) and even (fourth-order) terms is investigated in detail. The equation for the one-dimensional magnetic systems is integrable and admits exact solutions. It is more accurate than the nonlinear Schrödinger equation in describing wave propagation in the ocean and optical fibers. First, the modulation instability of solutions is analyzed in the presence of small perturbation. Second, breather and rogue wave solutions of this equation are constructed via the modified Darboux transformation method. The effects of the higher-order terms are investigated graphically. Specially, the interactions between two breathers are studied by adjusting the spectral parameters and the collisions between breather and rogue waves are also discussed.

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