Solutions of mKdV in classes of functions unbounded at infinity

Using P. Lax's concept of a Lax pair we prove 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 in particular, may tend to infinity with respect to the space variable. Moreover, we establish the invariance of the spectrum and the unitary type of the Schrodinger operator 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.