Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models

Nabile Boussïd; Andrew Comech

doi:10.3934/cpaa.2018065

Article Contents

2018, Volume 17, Issue 4: 1331-1347. Doi: 10.3934/cpaa.2018065

This issue Previous Article Singularity formation for the 1-D cubic NLS and the Schrödinger map on $\mathbb S^2$ Next Article Small amplitude solitary waves in the Dirac-Maxwell system

Spectral stability of bi-frequency solitary waves in Soler and Dirac-Klein-Gordon models

Nabile Boussïd^1, and
Andrew Comech^2,3,4, ,

1.
Universite Bourgogne Franche-Comté, 16 route de Gray, 25030 Besançon CEDEX, France
2.
Texas A & M University, College Station, TX 77843, USA
3.
IITP, Moscow 127051, Russia
4.
St. Petersburg State University, St. Petersburg 199178, Russia

* Corresponding author: Andrew Comech
* Corresponding author: Andrew Comech

Received: November 2017

Revised: February 2018

Published: April 2018

The first author aknowledges the support of Region Bourgogne Franche-Comté through the project "Projet du LMB: Analyse mathematique et simulation numérique d'EDP issues de problèmes de contrôle et du trafic routier". The second author was supported by the Russian Foundation for Sciences (project 14-50-00150).

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Abstract

We construct bi-frequency solitary waves of the nonlinear Dirac equation with the scalar self-interaction, known as the Soler model (with an arbitrary nonlinearity and in arbitrary dimension) and the Dirac-Klein-Gordon with Yukawa self-interaction. These solitary waves provide a natural implementation of qubit and qudit states in the theory of quantum computing.

We show the relation of $± 2ω\mathrm{i}$ eigenvalues of the linearization at a solitary wave, Bogoliubov $\mathbf{SU}(1,1)$ symmetry, and the existence of bi-frequency solitary waves. We show that the spectral stability of these waves reduces to spectral stability of usual (one-frequency) solitary waves.

Keywords:

Mathematics Subject Classification: Primary: 35C08, 35Q41, 37K40, 81Q05; Secondary: 37N20.

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