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

For each λ > 0 and under necessary conditions on the parameters, we construct normalized waves for second order PDE’s with mixed power non-linearities, with ‖ u ‖ L 2 ( R n ) 2 = λ , n ≥ 1 . 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 L 2 norm of the waves, namely the map λ → ω λ , 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 λ , the map λ → f ω λ is differentiable and the waves e i ω λ t f ω λ 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.