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

• DOI: 10.1007/s11071-017-3579-x
• Publication Date: 2017-07-17T05:40:11Z
• Authors: Qiuxia Xing; Zhiwei Wu; Dumitru Mihalache; Jingsong He
• Keywords: Eigenfunction; Dispersionless equation; Lax pair

The $n$-fold Darboux transformation $T_{n}$ of the focusing real mo\-di\-fied Kor\-te\-weg-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 $��_{j}$ 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 $��_{j}\rightarrow��_{1}$, 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.<br/>17 pages, 7 figures. This is the accepted version by Nonlinear Dynamics<br/>