Verify spectral assumptions for Petviashvili iteration in the semi-discrete Burgers traveling-wave integral equation

Determine whether the spectral assumptions required by Pelinovsky and Stepanyants (2004) for local convergence or divergence of Petviashvili iteration hold for the linearization at an exact traveling-wave profile of the nonlinear integral equation -c f(x) + (1/4) ∫_{x-1}^{x+1} f(z)^2 dz = C0 that characterizes traveling waves of the semi-discrete Burgers system 4 ū̇j + u^2_{j+1} - u^2_{j-1} = 0. Specifically, ascertain the spectrum of the linearized operator around an exact profile f for this integral equation and verify whether the requisite spectral properties are satisfied to justify Petviashvili iteration in this setting.

Background

The paper studies traveling-wave profiles for the semi-discrete Burgers system using dual variational formulations derived from a differential-difference equation (DDE) and an equivalent nonlinear integral equation (NIE). Petviashvili iteration is a standard numerical method for computing steady wave profiles of nonlinear dispersive equations, but its rigorous convergence theory relies on spectral assumptions about the linearized operator at the wave profile.

The authors note that while local convergence criteria by Pelinovsky and Stepanyants exist, these criteria depend on spectral properties that may not hold in non-integrable, nonlocal settings like the semi-discrete Burgers traveling-wave NIE. Clarifying whether these assumptions apply here would substantiate the use of Petviashvili iteration for this problem.