Stability Analysis of Breathers for Coupled Nonlinear Schrödinger Equations

The paper’s Theorem 2 claims uniform-in-time H^2-orbital stability to a fixed-(a,b1,b2) breather manifold while only allowing the scattering parameters c_{ij}(t) to vary with bounded time-derivatives. However, the CNLS/Manakov IST admits exact two-soliton solutions with slightly different real parts of the discrete eigenvalues λ1=(a+ε)+i b1 and λ2=(a−ε)+i b2. Such a solution is O(ε)-close at t=0 to the target breather q[2](·,0;a,b1,b2;c(0)) (since a enters via a Galilean phase T(a) and the profile is exponentially localized), but for times t≈const·ε^{-1} it splits into two pulses separated by O(1/ε) in the co-moving frame, so its H^2-distance to any member of the fixed-(a,b1,b2) breather family is bounded below by a positive constant. The theorem as stated does not constrain the perturbed initial data’s discrete spectrum to share a common real part a, nor does it allow a,b1,b2 to modulate; hence it is false. The model’s counterexample and proposed fixes (spectral alignment or modulation of spectral parameters) are correct. See the theorem statement and breather definition in the paper, which fix (a,b1,b2) and only vary c_{ij}(t) with a derivative bound , together with the Galilean transform used in the construction and the Darboux/IST framework that allows general two-soliton eigenvalues .