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 λj → λ 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 λ 1 → λ 0, where λ 0 is a special eigenvalue associated with the eigenfunction ϕ of the Lax pair of the mKdV equation. This eigenvalue λ 0, for which ϕ ( λ 0 ) = 0 , corresponds to the limit of infinite period of the periodic solution. 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.