Solutions of mKdV in classes of functions unbounded at infinity

We investigate the relation between the Korteweg de Vries and modified Korteweg de Vries equations (KdV and mKdV), and find a new algebro-analytic mechanism, similar to the Lax L-A pair, which involves a family of first-order operators Qλ depending on a spectral parameter λ, instead of the third-order operator A. In our framework, any generalized eigenfunction of the Schrödinger operator L, whose time-dependent potential solves the KdV equation, evolves according to a linear first-order partial differential equation, depending on the spectral parameter. This provides an explicit control over the time evolution. As an application, we establish global existence and uniqueness for solutions of the initial value problem for mKdV in classes of smooth functions which can be unbounded at infinity, and may even include functions which tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of L under the KdV flow and the invariance of the spectrum and the unitary type of the impedance operator under the mKdV flow for potentials in these classes.