Sharp Upper Bound for Amplitudes of Finite-Gap Solutions of the Modified Korteweg-de Vries Equation

Abstract

A sharp upper bound is established for the amplitudes of finite-gap solutions of the focusing modified Korteweg–de Vries equation. The proof shows that the optimization can be carried out entirely within the finite-dimensional hierarchy, without explicit integration of the finite-gap solution. The maximal amplitude is expressed in terms of the roots of the invariant polynomial associated with the finite-gap solution. An analogous formula is established for a bounded class of solutions of the defocusing modified Korteweg–de Vries equation. Both upper bounds are sharp and are attained by explicit initial data.