On the Existence and Stability of Normalized Ground States of the Kawahara, Fourth Order NLS and the Ostrovsky Equations

Abstract

In this dissertation we show the existence and stability of the normalized ground states for the Kawahara, fourth order nonlinear Schrödinger (NLS) and the generalized Ostrovsky equations. One of the starting points in our investigation were numerical stability results by S. Levandosky in [32], [31] which agree with our rigorous stability results. We show existence of the waves using variational techniques together with the concentration compactness argument. On the level of construction, we encounter certain obstacles in the form of new Gagliardo–Nirenberg–Sobolev type inequalities, which impose restrictions on the parameter space. We show stability utilizing spectral theory developed in the recent work by Z.Lin and C.Zeng in [35]. For the Kawahara model, our results provide a significant extension in the parameter space of the current rigorous results. In fact, our results rigorously establish the spectral stability for all acceptable values of the parameters. 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 the fourth order NLS equations 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 order derivatives vs pure Laplacian) has implications on the range of the existence and stability of the normalized waves. For the Ostrovsky equation, we show that all normalized waves we construct are spectrally stable. We also establish decay rates for the waves, extending the results in the paper by P. Zhang and Y. Liu [51].