Maximal amplitudes of finite-gap solutions for the focusing nonlinear Schrödinger equation (Q2406833)

Motivated by the understanding of exceptional waves (rogue waves) the authors provide an estimate of the modulus of specific Abelian functions in term of the sum of the imaginary part of the branchpoints of the related Riemann surface. These specific Abelian functions are quasi-periodic solutions of the focusing NLS equation (finite-gap solutions). They have been the object of the theory of (quasi)periodic inverse transform scattering. The proof of the inequality is by studying the critical points of such an holomorphic Abelian function \(f\) which is defined on \(\mathbb{T}^g\) (\(g\)-dimensional torus) with values in \(\mathbb{C}\). Using the symmetry properties of the Theta function the authors show that the critical points are located in \(\frac{1}{2}\mathbb{Z}^g\). The origin is shown to be the unique maximum of \(|f|\) and the main inequality of the paper follows. Besides, a new (simpler) proof of the positiveness of the Theta function in \(\mathbb{R}^g\) is provided in an appendix.