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

Iurii Posukhovskyi; Atanas G. Stefanov

doi:10.3934/dcds.2020175

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2020, Volume 40, Issue 7: 4131-4162. Doi: 10.3934/dcds.2020175

This issue Previous Article Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities Next Article Synchronisation of almost all trajectories of a random dynamical system

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

Iurii Posukhovskyi and
Atanas G. Stefanov^,

1460, Jayhawk Blvd., Department of Mathematics, University of Kansas, Lawrence, KS 66049, USA

^* Corresponding author: Atanas G. Stefanov
^* Corresponding author: Atanas G. Stefanov

Received: February 2019

Revised: November 2019

Published: April 2020

The first author was partially supported as a graduate research assistant from grant NSF-DMS 1614734. The second author is partially supported by NSF-DMS 1908626.

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Abstract

We consider the Kawahara model and two fourth order semi-linear Schrödinger 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. In fact, our results establish (modulo an additional technical assumption, which should be satisfied at least generically), spectral stability for all normalized waves constructed therein - in all dimensions, for all acceptable values of the parameters. This, combined with the results of [5], provides orbital stability, for all normalized waves enjoying the non-degeneracy property. The validity of the non-degeneracy property for generic waves remains an intriguing open question.

At the same time, we verify and clarify recent numerical simulations of the spectral stability of these solitons. For the fourth order NLS models, we improve upon recent results on spectral stability of very special, explicit solutions in the one dimensional case. Our multidimensional results for fourth order anisotropic 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.

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Mathematics Subject Classification: Primary: 35Q55, 35Q51; Secondary: 35G16.

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