Instabilities via negative Krein signature in a weakly non-Hamiltonian DNLS model

Panayotis G. Kevrekidis; Panayotis G. Kevrekidis

doi:10.3934/mine.2019.2.378

Mathematics in Engineering

2019, Volume 1, Issue 2: 378-390. doi: 10.3934/mine.2019.2.378

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Instabilities via negative Krein signature in a weakly non-Hamiltonian DNLS model

Panayotis G. Kevrekidis ^,

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA

Received: 18 January 2019 Accepted: 27 March 2019 Published: 26 April 2019

In the present work we consider a model that has been proposed at the continuum level for self-defocusing nonlinearities in atomic Bose-Einstein condensates (BECs) in order to capture phenomenologically the loss of condensate atoms to thermal ones. We explore a model combining dispersion, nonlinearity and gain/loss at the discrete level, and illustrate the idea that modes associated with negative "energy" (mathematically: negative Krein signature) can give rise to instability of excited states when non-Hamiltonian terms are introduced in a nonlinear dynamical lattice. We showcase this idea by considering one-, two- and three-site discrete modes, exploring their stability via analytical approximations, and corroborating their continuation numerically over the relevant parameter controlling the strength of the weakly non-Hamiltonian term. We also manifest through direct numerical simulations their unstable nonlinear dynamics.
- solitary waves,
- instabilities,
- Krein signature,
- DNLS model
Citation: Panayotis G. Kevrekidis. Instabilities via negative Krein signature in a weakly non-Hamiltonian DNLS model[J]. Mathematics in Engineering, 2019, 1(2): 378-390. doi: 10.3934/mine.2019.2.378

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Abstract

In the present work we consider a model that has been proposed at the continuum level for self-defocusing nonlinearities in atomic Bose-Einstein condensates (BECs) in order to capture phenomenologically the loss of condensate atoms to thermal ones. We explore a model combining dispersion, nonlinearity and gain/loss at the discrete level, and illustrate the idea that modes associated with negative "energy" (mathematically: negative Krein signature) can give rise to instability of excited states when non-Hamiltonian terms are introduced in a nonlinear dynamical lattice. We showcase this idea by considering one-, two- and three-site discrete modes, exploring their stability via analytical approximations, and corroborating their continuation numerically over the relevant parameter controlling the strength of the weakly non-Hamiltonian term. We also manifest through direct numerical simulations their unstable nonlinear dynamics.

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