Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation

Pramana: Journal of physics

YONG CHEN^2,3¹  XIAO-YONG WEN^1² 

[1] Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing 100190, China^2;School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China^1;School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China^3

关键词: Generalised perturbation ($n;    N − n$)-fold Darboux transformation;    mKdV equation;    rational soliton solutions;    numerical simulations;   

DOI  :  

学科分类：物理（综合）

来源: Indian Academy of Sciences

【 摘 要 】

In this paper, we propose a generalised perturbation ($n, N − n$)-fold Darboux transformation (DT) of the modified Korteweg–de Vries (mKdV) equation using the Taylor expansion and a parameter limit procedure. We apply the generalised perturbation ($1, N − 1$)-fold DT to find the new explicit higher-order rational soliton (RS) solutions in terms of determinants of the mKdV equation. These higher-order RS solutions are different from those known soliton results in terms of hyperbolic functions which are obtained from the classical iterated DT. The dynamics behaviours of the first-, second-, third-, and fourth-order RS solutions are shown graphically. The wave propagation characteristics and stability are also discussed using numerical simulations. We find that the initial constant seed solution plays an important role on the wave propagation stability of RS. Through Miura transformation, we give some complex higher-order rational solutions of the Korteweg–de Vries (KdV) equation which are different from the known results. The relevant structures also are discussed using some figures. The method used can also be extended to seek explicit rational solutions of other nonlinear integrable equations.

【 授权许可】

CC BY