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

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 $lambda_{j}rightarrowlambda_{1}$, 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.

Construction of rational solutions of the real modified Korteweg-de Vries equation from its periodic solutions

In this paper, we consider the real modified Korteweg-de Vries (mKdV) equation and construct a special kind of breather solution, which can be obtained by taking the limit $lambda_{j}$ $rightarrow$ $lambda_{1}$ of the Lax pair eigenvalues used in the $n$-fold Darboux transformation that generates the order-$n$ periodic solution from a constant seed solution. Further, this special kind of breather solution of order $n$ can be used to generate the order-$n$ rational solution by taking the limit $lambda_{1}$ $rightarrow$ $lambda_{0}$, where $lambda_{0}$ is a special eigenvalue associated to the eigenfunction $phi$ of the Lax pair of the mKdV equation. This eigenvalue $lambda_0$, for which $phi(lambda_0)=0$, corresponds to the limit of infinite period of the periodic solution. %This second limit of double eigenvalue degeneration might be realized approximately in optical fibers, in which an injected %initial ideal pulse is created by a comb system and a programmable optical filter according to the profile of the analytical %form of the b-positon at a certain spatial position $x_{0}$. Therefore, we suggest a new way to observe the higher-order %rational solutions in optical fibers, namely, to measure the wave patterns at the central region of the higher order b-positon %generated by ideal initial pulses when the eigenvalue $lambda_{1}$ is approaching $lambda_{0}$. Our analytical and numerical results show the effective mechanism of generation of higher-order rational solutions of the mKdV equation from the double eigenvalue degeneration process of multi-periodic solutions.