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.