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We study the modulational stability of periodic travelling wave solutions toequations of nonlinear Schr\"odinger type. In particular, we prove that thecharacteristics of the quasi-linear system of equations resulting from a slowmodulation approximation satisfy the same equation, up to a change invariables, as the normal form of the linearized spectrum crossing the origin.This normal form is taken from arXiv:1910.05392, where Leisman et al. computethe spectrum of the linearized operator near the origin via an analysis ofJordan chains. We derive the modulation equations using Whitham's formalmodulation theory, in particular the variational principle applied to anaveraged Lagrangian. We use the genericity conditions assumed in the rigoroustheory of Leisman et al. to direct the homogenization of the modulationequations. As a result of the agreement between the equation for thecharacteristics and the normal form from the linear theory, we show that thehyperbolicity of the Whitham system is a necessary condition for modulationalstability of the underlying wave.

Rigorous justification of the Whitham modulation theory for equations of NLS type