Numerical instability of the Akhmediev breather and a finite-gap model of it (Q2275145)

This paper provides numerical illustrations of the authors' previous work about the finite gap method used to find space periodic solutions to the nonlinear Schrödinger (NLS) equation. Let us explain the framework before commenting on the authors' results. If the initial data is a small harmonic perturbation of the constant 1 and the period of the perturbation belongs to \(]\pi,2\pi[\) then the solution of the NLS equation is modulationally unstable: it concentrates periodically in time according to Akhmediev breathers, the so-called rogue waves. The numerical calculations raise one unexpected difficulty: round-off errors. Indeed, the authors need to make calculations in quadruple precisions to validate the theoretical behavior previously described. Otherwise even in double precision the numerical solution does not reproduce the time periodicity. This phenomenon is even stronger as the time step decreases. Under quadruple precision the numerical results validate the periodical behaviour of rogue wave occurence and the authors found an empirical formula relating this time recurrence of rogue waves and the numerical time step. Moreover, the numerical results also validate the fact that the modulational instability is preserved when perturbating the initial data by slow harmonic waves (with wavelength bigger than \(2\pi\)) as predicted by the theory. For the entire collection see [Zbl 1412.37003].