Complexitons of the modified KdV equation by Darboux transformation

2008 ◽  
Vol 196 (2) ◽  
pp. 501-510 ◽  
Author(s):  
Hongxia Wu ◽  
Yunbo Zeng ◽  
Tianyou Fan
Keyword(s):  
Darboux Transformation ◽  
Kdv Equation ◽  
Modified Kdv Equation

scholarly journals Rational Solutions of Multi-component Nonlinear Schrödinger Equation and Complex Modified KdV Equation

2021 ◽  
Author(s):  
Lihong Wang ◽  
Jingsong He ◽  
Róbert Erdélyi
Keyword(s):  
Explicit Formula ◽  
Darboux Transformation ◽  
Critical Condition ◽  
Vries Equation ◽  
Kdv Equation ◽  
Rational Solution ◽  
Rogue Waves ◽  
Rational Solutions ◽  
Modified Kdv Equation ◽  
De Vries

In this paper, the critical condition to achieve rational solutions of the multi-component nonlinear Schr\”odinger equation is proposed by introducing two nilpotent Lax matrices. Taking the series multisections of the vector eigenfunction as a set of fundamental eigenfunctions,an explicit formula of the $n$th-order rational solution is obtained by the degenerate Darboux transformation, which is used to generate some new patterns of rogue waves. A conjecture about the degree of the $n$th-order rogue waves is summarized. This conjecture also holds for rogue waves of the multi-component complex modified Korteweg-de Vries equation. Finally, the semi-rational solutions of the Manakov system are discussed.

Higher-order rational soliton solutions for the fifth-order modified KdV and KdV equations

2021 ◽  
pp. 2150036
Author(s):  
Zhi-Jie Pei ◽  
Hai-Qiang Zhang
Keyword(s):  
Darboux Transformation ◽  
Fourth Order ◽  
Taylor Expansion ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Kdv Equations ◽  
Perturbation Formula ◽  
Modified Kdv Equation ◽  
Fifth Order

In this paper, we construct the generalized perturbation ([Formula: see text], [Formula: see text])-fold Darboux transformation of the fifth-order modified Korteweg-de Vries (KdV) equation by the Taylor expansion. We use this transformation to derive the higher-order rational soliton solutions of the fifth-order modified KdV equation. We find that these higher-order rational solitons admit abundant interaction structures. We graphically present the dynamics behaviors from the first- to fourth-order rational solitons. Furthermore, by the Miura transformation, we obtain the complex rational soliton solutions of the fifth-order KdV equation.

scholarly journals Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

2012 ◽  
Vol 45 (4) ◽  
pp. 045206 ◽  
Author(s):  
Jun-ichi Inoguchi ◽  
Kenji Kajiwara ◽  
Nozomu Matsuura ◽  
Yasuhiro Ohta
Keyword(s):  
Kdv Equation ◽  
Plane Curves ◽  
Explicit Solutions ◽  
Modified Kdv Equation
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