On the normalized ground states for the Kawahara equation and a fourth order NLS

TL;DR

This paper constructs and analyzes the spectral stability of normalized ground states for the Kawahara equation and fourth order nonlinear Schrödinger equations across multiple dimensions, extending existing results and revealing new stability paradigms.

Contribution

It provides the first multidimensional stability results for fourth order NLS and significantly extends the parameter range for Kawahara ground states, introducing a new stability paradigm based on second derivative forms.

Findings

Spectrally stable normalized ground states for Kawahara and fourth order NLS.

Extended parameter space for Kawahara ground state stability.

Discovery of a new paradigm linking second derivative form to stability and existence.

Abstract

We consider the Kawahara model and two fourth order semi-linear Schr\"odinger equations in any spatial dimension. We construct the corresponding normalized ground states, which we rigorously show to be spectrally stable. For the Kawahara model, our results provide a significant extension in parameter space of the current rigorous results. At the same time, we verify and clarify recent numerical simulations of the stability of these solitons. For the fourth order NLS models, we improve upon recent results on stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order NLS seem to be the first of its kind. Of particular interest is a new paradigm that we discover herein. Namely, all else being equal, the form of the second order derivatives (mixed second derivatives vs. pure Laplacian) has implications on the range of existence and stability of the normalized waves.