Rigorous stability proof for DNSE breather solutions across all nonlinearity strengths

Establish a rigorous analytic proof that, for the discrete nonlinear Schrödinger equation \dot{c}_n(t) = i(2c_n(t) - c_{n-1}(t) - c_{n+1}(t) - \alpha |c_n(t)|^{2}c_n(t)) on a finite periodic lattice, the unimodal site-centered breather solutions c_n(t)=e^{i\Omega t} r_n (with r_n solving the nonlinear eigenvalue problem (2.6)) are linearly stable for every nonzero \alpha by proving that the spectrum of the linearized stability operator (2.9) consists solely of purely imaginary eigenvalues over the entire \alpha-range. Complementarily, ascertain the linear instability of bond-centered breathers by demonstrating the existence of eigenvalue pairs with nonzero real parts across all \alpha.

Background

The paper studies breather (spatially localized, time-dependent) solutions in the discrete nonlinear Schrödinger equation (DNSE) and analyzes their linear stability via a 2L×2L eigenvalue problem (equation (2.9)). Numerical evidence shows that unimodal site-centered breathers have purely imaginary eigenvalues and are linearly stable (though not asymptotically), while bond-centered breathers possess eigenvalue pairs with nonzero real parts and are linearly unstable.

Although these behaviors are widely recognized in the literature and supported by numerical computations, the authors explicitly note the absence of a formal analytic proof that is valid uniformly for all values of the nonlinearity parameter \alpha. The challenge spans regimes from the anti-integrable limit (large \alpha) to the continuum limit (small \alpha), and a unified, rigorous treatment across the entire \alpha-range has not yet been established.