On the Normalized Ground States of Second Order PDE's with Mixed Power Non-linearities

For each and under necessary conditions on the parameters, we construct normalized waves for second order PDE's with mixed power non-linearities, with . We show that these are bell-shaped smooth and localized functions, and we compute their precise asymptotics. We study the question for the smoothness of the Lagrange multiplier with respect to the norm of the waves, namely the map , a classical problem related to its stability. We show that this is intimately related to the question for the non-degeneracy of the said solitons. We provide a wide class of non-linearities, for which the waves are non-degenerate. Under some minimal extra assumptions, we show that a.e. in , the map is differentiable and the waves are spectrally (and in some cases orbitally) stable as solutions to the NLS equation. Similar results are obtained for the same waves, as traveling waves of the Zakharov-Kuznetsov system.