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

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Abstract

The n-fold Darboux transformation \(T_{n}\) 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 \(\lambda _{j}\) 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 \(\lambda _{j}\rightarrow \lambda _{1}\), 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.

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Acknowledgements

This work is supported by the NSF of China Under Grant No. 11671219, and the K. C. Wong Magna Fund in Ningbo University. We thank members of our group at Ningbo University for useful discussions on this manuscript.

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  1. Department of Mathematics, Ningbo University, Ningbo, 315211, Zhejiang, People’s Republic of China

    Qiuxia Xing, Zhiwei Wu & Jingsong He

  2. Horia Hulubei National Institute for Physics and Nuclear Engineering, P.O.Box MG–6, 077125, Magurele, Romania

    Dumitru Mihalache

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Xing, Q., Wu, Z., Mihalache, D. et al. Smooth positon solutions of the focusing modified Korteweg–de Vries equation. Nonlinear Dyn 89, 2299–2310 (2017). https://doi.org/10.1007/s11071-017-3579-x

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