Modulational instability for nonlinear Schrödinger equations with a periodic potential (Q2504347)

The authors deal with the stability of standing wave solutions to the nonlinear Schrödinger equation (NLS) with an external potential: \[ i\psi_t= -\tfrac12\,\psi_{xx}\pm \varepsilon|\psi|^2\psi+ V(x)\,\psi,\tag{1} \] where the potential \(V(x)\) has period 1. They find that the combination of the Hamiltonian structure and the \(\mathbb{U}(1)\) symmetry dramatically simplifies the structure and possible bifurcation of the spectrum of the linearized operator. The authors find a simple sufficient condition which guarantees the existence of a modulation instability. In the case of weakly nonlinear standing waves, when \(\varepsilon\) is small, they apply a perturbation argument to show that the lower bound edges are modulationally unstable in the focusing case, while the upper bound edges are modulationally unstable in the defocusing case.