Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

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

Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-4515, USA
$^\dagger$This contribution is part of the Special Issues: Hamiltonian Lattice Dynamics
  Guest Editors: Simone Paleari; Tiziano Penati
  Link: http://www.aimspress.com/newsinfo/1165.html

Special Issues: Hamiltonian Lattice Dynamics

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.
  Article Metrics

Keywords solitary waves; instabilities; Krein signature; DNLS model

Citation: Panayotis G. Kevrekidis. Instabilities via negative Krein signature in a weakly non-Hamiltonian DNLS model. Mathematics in Engineering, 2019, 1(2): 378-390. doi: 10.3934/mine.2019.2.378


  • 1. Kevrekidis PG (2009) The Discrete Nonlinear Schrödinger Equation: Mathematical Analysis, Numerical Computation and Physical Perspectives, Heidelberg: Springer-Verlag.
  • 2. Christodoulides DN, Lederer F, Silberberg Y, Christodoulides DN (2003) Discretizing light behavior in linear and nonlinear waveguide lattices. Nature 424: 817–823.    
  • 3. Sukhorukov AA, Kivshar YS, Eisenberg HS, et al. (2003) Spatial optical solitons in waveguide arrays. IEEE J Quantum Elect 39: 31–50.    
  • 4. Lederer F, Stegeman GI, Christodoulides DN, et al. (2008) Discrete solitons in optics. Phys Rep 463: 1–126.    
  • 5. Morsch O, Oberthaler M (2006) Dynamics of Bose-Einstein condensates in optical lattices. Rev Mod Phys 78: 179–215.    
  • 6. Eisenberg HS, Silberberg Y, Morandotti R, et al. (1998) Discrete spatial optical solitons in waveguide arrays. Phys Rev Lett 81: 3383–3386.    
  • 7. Eisenberg HS, Silberberg Y, Morandotti R, et al. (2000) Diffraction management. Phys Rev Lett 85: 1863–1866.    
  • 8. Morandotti R, Peschel U, Aitchison JS, et al. (1999) Dynamics of discrete solitons in optical waveguide arrays. Phys Rev Lett 83: 2726–2729.    
  • 9. Morandotti R, Eisenberg HS, Silberberg Y, et al. (2001) Self-focusing and defocusing in waveguide arrays. Phys Rev Lett 86: 3296–3299.    
  • 10. Neshev DN, Alexander TJ, Ostrovskaya EA, et al. (2004) Observation of discrete vortex solitons in optical induced photonic lattices. Phys Rev Lett 92: 123903.    
  • 11. Fleischer JW, Bartal G, Cohen O, et al. (2004) Observation of vortex-ring "discrete" solitons in 2D photonic lattices. Phys Rev Lett 92: 123904.    
  • 12. Iwanow R, May-Arrioja DA, Christodoulides DN, et al. (2005) Discrete Talbot effect in waveguide arrays. Phys Rev Lett 95: 053902.    
  • 13. Rüter CE, Makris KG, El-Ganainy R, et al. (2010) Observation of parity-time symmetry in optics. Nat Phys 6: 192–195.    
  • 14. Pitaevskii LP, Stringari S (2003) Bose-Einstein Condensation, Oxford: Oxford University Press.
  • 15. Kevrekidis PG, Frantzeskakis DJ, Carretero-González R (2015) The Defocusing Nonlinear Schrödinger Equation, Philadelphia: SIAM.
  • 16. Proukakis N, Gardiner S, Davis M, et al. (2013) Quantum gases: Finite Temperature and Nonequilibrium Dynamics, London: Imperial College Press.
  • 17. Carr LD (2010) Understanding Quantum Phase Transitions, Boca Raton: CRC Press Taylor & Francis Group.
  • 18. Pitaevskii LP (1959) Phenomenological theory of superfluidity near the λ point. Sov Phys JETP 35: 282–287.
  • 19. Jackson B, Proukakis NP (2008) Finite temperature models of Bose-Einstein condensation. J Phys B: At Mol Opt Phys 41: 203002.
  • 20. Cockburn SP, Nistazakis HE, Horikis TP, et al. (2010) Matter wave dark solitons: Stochastic versus analytical results. Phys Rev Lett 104: 174101.    
  • 21. Cockburn SP, Proukakis NP (2009) The stochastic Gross-Pitaevskii and some applications. Laser Phys 19: 558–570.    
  • 22. Kevrekidis PG, Frantzeskakis DJ (2011) Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete Contin Dyn Sys 4: 1199–1212.
  • 23. Achilleos V, Yan D, Kevrekidis PG, et al. (2012) Dark-bright solitons in Bose-Einstein condensates at finite temperatures. New J Phys 14: 055006.    
  • 24. Yan D, Carretero-González R, Frantzeskakis DJ, et al. (2014) Exploring vortex dynamics in the presence of dissipation: analytical and numerical results. Phys Rev A 89: 043613.    
  • 25. Moon G, Kwon WJ, Lee H, et al. (2015) Thermal friction on quantum vortices in a Bose-Einstein condensate. Phys Rev A 92: 051601.    
  • 26. Kevrekidis PG, Susanto H, Chen Z (2006) Higher-order-mode soliton structures in two- dimensional lattices with defocusing nonlinearity. Phys Rev E 74: 066606.    
  • 27. Pelinovsky DE, Kevrekidis PG, Frantzeskakis DJ (2005) Stability of discrete solitons in nonlinear Schrödinger lattices. Phys D 212: 1–19.    
  • 28. Kapitula T, Kevrekidis PG, Sandstede B (2004) Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian systems. Phys D 195: 263–282.    
  • 29. Aranson IS, Kramer L (2002) The world of the complex Ginzburg-Landau equation. Rev Mod Phys 74: 99–143.    
  • 30. Efremidis NK, Christodoulides DN (2003) Discrete Ginzburg-Landau solitons. Phys Rev E 67: 026606.    
  • 31. Efremidis NK, Christodoulides DN, Hizanidis K (2007) Two-dimensional discrete Ginzburg- Landau solitons. Phys Rev A 76: 043839.    
  • 32. Karachalios NI, Nistazakis HE, Yannacopoulos AN (2007)Asymptotic behavior of solutions of complex discrete evolution equations: the discrete Ginzburg-Landau equation. Discrete Contin Dyn Sys-Ser A 19: 711–736.
  • 33. Kiselev Al S, Kiselev An S, Rozanov NN (2008) Dissipative discrete spatial optical solitons in a system of coupled optical fibers with the Kerr and resonance nonlinearities. Opt Spectrosc 105: 547–556.    
  • 34. MacKay RS, Aubry S (1994) Proof of existence of breathers for time-reversible or Hamiltonian networks of weakly coupled oscillators. Nonlinearity 7: 1623–1643.    
  • 35. Darmanyan S, Kobyakov A, Lederer F, et al. (1998) Stability of strongly localized excitations in discrete media with cubic nonlinearity. J Exp Theor Phys 86: 682–686.    
  • 36. Achilleos V, Bishop AR, Diamantidis S, et al. (2016) Dynamical playground of a higher-order cubic Ginzburg-Landau equation: From orbital connections and limit cycles to invariant tori and the onset of chaos. Phys Rev E 94: 012210.    


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved