Traveling waves for the Nonlinear Schrödinger Equation with general nonlinearity in dimension one

We study the traveling waves of the Nonlinear Schrödinger Equation in dimension one. Through various model cases, we show that for nonlinearities having the same qualitative behaviour as the standard Gross-Pitaevkii one, the traveling waves may have rather different properties. In particular, our examples exhibit multiplicity or nonexistence results, cusps (as for the Jones-Roberts curve in the three-dimensional Gross-Pitaevskii equation), and a transonic limit which can be the modified (KdV) solitons or even the generalized (KdV) soliton instead of the standard (KdV) soliton.