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

The n-fold Darboux transformation Tn 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) -> 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.