Instability of double-periodic waves in the nonlinear Schrödinger equation

The paper’s main claims and computational evidence align: both double-periodic families (2.15) and (2.16) are spectrally unstable via the Lax-pair/Floquet separation, with (2.15)'s unstable spectrum sitting on the strip boundary Im(Λ)=±2π/T and (2.16) exhibiting a figure-eight band plus boundary bands; amplitude-normalized rates are generally smaller than for standing periodic waves. These are all documented in the text and figures (Lax pair (2.5)–(2.6); mapping (4.1)–(4.3); boundary location for (2.15); figure-eight and boundary bands for (2.16); and the concluding comparison) . The candidate solution gets most of this framework right (Floquet–Lax dictionary, squared-eigenfunction mapping, existence of off-imaginary Lax spectrum, boundary placement for (2.15)), but it asserts an incorrect monotonic trend for the maximal growth rate of (2.16) as k→1: the paper shows the rate decreases and then increases again to a maximum as k→1, not a monotone decrease .