Sharp Upper Bound for Amplitudes of Hyperelliptic Solutions of the Focusing Nonlinear Schrödinger Equation

Hyperelliptic or finite-gap solutions of the focusing nonlinear Schrödinger equation are quasiperiodic N-phase solutions whose time dependence linearizes on a real subtorus of the Jacobi variety of an invariant hyperelliptic Riemann surface. A new proof is given for the formula for an upper bound, attained for some initial values, on amplitudes of hyperelliptic solutions. The upper bound is equal to the sum of the imaginary parts of all the branch points of the invariant Riemann surface which lie in the upper half-plane. A similar formula is proven for bounded solutions of the defocusing nonlinear Schrödinger equation. The new proof generalizes the derivation of the two-phase formula (Wright 2016 Physica D 321–2 16–38), and the result is the same N-phase formula obtained using Riemann–Hilbert methods by Bertola and Tovbis (2017 Commun. Math. Phys. 354 525–47)