  We consider reduced-order modeling of nonlinear dispersive waves described by
a class of nonlinear Schrodinger (NLS) equations. We compare two nonlinear
reduced-order modeling methods: (i) The reduced Lagrangian approach which
relies on the variational formulation of NLS and (ii) The recently developed
method of reduced-order nonlinear solutions (RONS). First, we prove the
surprising result that, although the two methods are seemingly quite different,
they can be obtained from the real and imaginary parts of a single
complex-valued master equation. Furthermore, for the NLS equation in a
stationary frame, we show that the reduced Lagrangian method fails to predict
the correct group velocity of the waves whereas RONS predicts the correct group
velocity. Finally, for the modified NLS equation, where the reduced Lagrangian
approach is inapplicable, the RONS reduced-order model accurately approximates
the true solutions.
