Multiprecision computation of bright and dark solitons in the discrete nonlinear Schrödinger equation
We study the spectral stability of bright and dark solitons in the discrete nonlinear Schrödinger (DNLS) equation using multiprecision arithmetic. The eigenvalues governing stability are exponentially small in the lattice spacing and cannot be resolved with standard double precision. To address this, we develop a computational framework combining multiprecision arithmetic, an exact Jacobian for the stationary problem, and a squared-operator formulation for spectral analysis. This enables accurate resolution of exponentially small eigenvalues and direct comparison with exponential-asymptotic predictions. Our results show that onsite bright solitons are spectrally stable, whereas intersite bright solitons and both onsite and intersite dark solitons are unstable. Bright solitons require only a few eigenvalues and allow efficient large-scale computations, while dark solitons demand higher precision due to their proximity to the continuous spectrum. Simulations up to \(N=65{,}250\) grid points (31.7 GB RAM) highlight the necessity of multiprecision arithmetic for capturing beyond-all-orders spectral effects.