Modulated amplitude waves with non-trivial phase in quasi-1D inhomogeneous Bose–Einstein condensates

•A rigorous proof of the existence of rotating MAWs in an inhomogeneous BEC.•No condition on the sign or the magnitude of the trap or the scattering length.•Non-trivial phase leads to a singular ODE for the amplitude.•The proof combines a local continuation theorem and properties of rotation numbers.

We consider a 1D nonlinear Schrödinger equation (NLSE) which describes the mean field dynamics of an elongated Bose–Einstein condensate and prove the existence of modulated amplitude waves with non-trivial phase and minimal spatial period tending to infinite. The proof combines the theory of local continuation of non-degenerate periodic solutions with a property of the Ermakov–Pinney equation.