  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.
