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

Abstract

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 ${\lambda }_j$ 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 ${\lambda }_j\to {\lambda }_1$, 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.

17 pages, 7 figures. This is the accepted version by Nonlinear Dynamics