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

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 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 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.