A new high-order spectral problem of the mKdV and its associated integrable decomposition

A new approach to formulizing a new high-order matrix spectralproblem from a normal 2× 2 matrix modified Korteweg--de Vries(mKdV) spectral problem is presented. It is found that theisospectral evolution equation hierarchy of this new higher-ordermatrix spectral problem turns out to be the well-known mKdV equationhierarchy. By using the binary nonlinearization method, a newintegrable decomposition of the mKdV equation is obtained in thesense of Liouville. The proof of the integrability shows thatr-matrix structure is very interesting.