# Smooth positon solutions of the focusing modified Korteweg–de Vries equation

Smooth positon solutions of the focusing modified Korteweg–de Vries equation The n-fold Darboux transformation $$T_{n}$$ T n of the focusing real modified Korteweg–de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues $$\lambda _{j}$$ λ j and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit $$\lambda _{j}\rightarrow \lambda _{1}$$ λ j → λ 1 , from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding “phase shifts” of the multi-positons are also investigated. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Nonlinear Dynamics Springer Journals

# Smooth positon solutions of the focusing modified Korteweg–de Vries equation

, Volume 89 (4) – Jul 17, 2017
12 pages

/lp/springer_journal/smooth-positon-solutions-of-the-focusing-modified-korteweg-de-vries-MVl4w9XHFq
Publisher
Springer Netherlands
Subject
Engineering; Vibration, Dynamical Systems, Control; Classical Mechanics; Mechanical Engineering; Automotive Engineering
ISSN
0924-090X
eISSN
1573-269X
D.O.I.
10.1007/s11071-017-3579-x
Publisher site
See Article on Publisher Site

### Abstract

The n-fold Darboux transformation $$T_{n}$$ T n of the focusing real modified Korteweg–de Vries (mKdV) equation is expressed in terms of the determinant representation. Using this representation, the n-soliton solutions of the mKdV equation are also expressed by determinants whose elements consist of the eigenvalues $$\lambda _{j}$$ λ j and the corresponding eigenfunctions of the associated Lax equation. The nonsingular n-positon solutions of the focusing mKdV equation are obtained in the special limit $$\lambda _{j}\rightarrow \lambda _{1}$$ λ j → λ 1 , from the corresponding n-soliton solutions and by using the associated higher-order Taylor expansion. Furthermore, the decomposition method of the n-positon solution into n single-soliton solutions, the trajectories, and the corresponding “phase shifts” of the multi-positons are also investigated.

### Journal

Nonlinear DynamicsSpringer Journals

Published: Jul 17, 2017

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