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

Abstract

A direct proof based on commuting finite-dimensional flows and local polynomial invariants is given for a sharp upper bound on the amplitudes of finite-gap solutions of the modified Korteweg-de Vries (mKdV) equation. The maximal amplitude is the sum of the imaginary parts of the upper-half-plane square roots of the roots of the invariant polynomial of the finite-gap solution of the focusing mKdV equation. An analogous formula is established for a bounded class of solutions of the defocusing mKdV equation. The upper bounds are sharp and are explicitly attained by suitable initial data.