Journal of High Energy Physics

, 2019:106 | Cite as

Stability analysis of classical string solutions and the dressing method

  • Dimitrios Katsinis
  • Ioannis Mitsoulas
  • Georgios PastrasEmail author
Open Access
Regular Article - Theoretical Physics


The dressing method is a technique to construct new solutions in non-linear sigma models under the provision of a seed solution. This is analogous to the use of autoBäcklund transformations for systems of the sine-Gordon type. In a recent work, this method was applied to the sigma model that describes string propagation on ℝ × S2, using as seeds the elliptic string solutions. Some of the new solutions that emerge reveal instabilities of their elliptic precursors [1]. The focus of the present work is the fruitful use of the dressing method in the study of the stability of closed string solutions. It establishes an equivalence between the dressing method and the conventional linear stability analysis. More importantly, this equivalence holds true in the presence of appropriate periodicity conditions that closed strings must obey. Our investigations point to the direction of the dressing method being a general tool for the study of the stability of classical string configurations in the diverse class of symmetric spacetimes.


Bosonic Strings Long strings 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    D. Katsinis, I. Mitsoulas and G. Pastras, Salient Features of Dressed Elliptic String Solutions on ℝ × S 2, arXiv:1903.01408 [INSPIRE].
  2. [2]
    M. Spradlin and A. Volovich, Dressing the Giant Magnon, JHEP 10 (2006) 012 [hep-th/0607009] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  3. [3]
    C. Kalousios, M. Spradlin and A. Volovich, Dressing the giant magnon II, JHEP 03 (2007) 020 [hep-th/0611033] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    D. Katsinis, I. Mitsoulas and G. Pastras, Elliptic string solutions on ℝ × S 2 and their pohlmeyer reduction, Eur. Phys. J. C 78 (2018) 977 [arXiv:1805.09301] [INSPIRE].
  5. [5]
    S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A Semiclassical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99 [hep-th/0204051] [INSPIRE].
  6. [6]
    D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
  7. [7]
    D.M. Hofman and J.M. Maldacena, Giant Magnons, J. Phys. A 39 (2006) 13095 [hep-th/0604135] [INSPIRE].
  8. [8]
    R. Ishizeki and M. Kruczenski, Single spike solutions for strings on S 2 and S 3, Phys. Rev. D 76 (2007) 126006 [arXiv:0705.2429] [INSPIRE].
  9. [9]
    K. Okamura and R. Suzuki, A Perspective on Classical Strings from Complex sine-Gordon Solitons, Phys. Rev. D 75 (2007) 046001 [hep-th/0609026] [INSPIRE].
  10. [10]
    A.E. Mosaffa and B. Safarzadeh, Dual spikes: New spiky string solutions, JHEP 08 (2007) 017 [arXiv:0705.3131] [INSPIRE].
  11. [11]
    B.-H. Lee and C. Park, Unbounded Multi Magnon and Spike, J. Korean Phys. Soc. 57 (2010) 30 [arXiv:0812.2727] [INSPIRE].
  12. [12]
    M. Kruczenski, J. Russo and A.A. Tseytlin, Spiky strings and giant magnons on S 5, JHEP 10 (2006) 002 [hep-th/0607044] [INSPIRE].
  13. [13]
    K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].
  14. [14]
    V.E. Zakharov and A.V. Mikhailov, Relativistically Invariant Two-Dimensional Models in Field Theory Integrable by the Inverse Problem Technique (in Russian), Sov. Phys. JETP 47 (1978) 1017 [INSPIRE].
  15. [15]
    V.E. Zakharov and A.V. Mikhailov, On the integrability of classical spinor models in two-dimensional space-time, Commun. Math. Phys. 74 (1980) 21 [INSPIRE].
  16. [16]
    J.P. Harnad, Y. Saint Aubin and S. Shnider, Backlund Transformations for Nonlinear σ Models With Values in Riemannian Symmetric Spaces, Commun. Math. Phys. 92 (1984) 329 [INSPIRE].
  17. [17]
    T.J. Hollowood and J.L. Miramontes, Magnons, their Solitonic Avatars and the Pohlmeyer Reduction, JHEP 04 (2009) 060 [arXiv:0902.2405] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    D. Katsinis, I. Mitsoulas and G. Pastras, Dressed elliptic string solutions on ℝ × S 2, Eur. Phys. J. C 78 (2018) 668 [arXiv:1806.07730] [INSPIRE].
  19. [19]
    C.K.R.T. Jones, R. Marangell, P.D. Miller, R.G. Plaza, On the Stability Analysis of Periodic Sine-Gordon Traveling Waves, Physica D 251 (2013) 63 [arXiv:1210.0659].
  20. [20]
    I. Bakas and G. Pastras, On elliptic string solutions in AdS 3 and dS 3, JHEP 07 (2016) 070 [arXiv:1605.03920] [INSPIRE].
  21. [21]
    G. Pastras, Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics, 2017, arXiv:1706.07371 [INSPIRE].
  22. [22]
    B. Wang, Stability of Helicoids in Hyperbolic Three-Dimensional Space, arXiv:1502.04764.
  23. [23]
    B. Wang, Least Area Spherical Catenoids in Hyperbolic Three-Dimensional Space, arXiv:1204.4943.
  24. [24]
    G. Pastras, Static elliptic minimal surfaces in AdS 4, Eur. Phys. J. C 77 (2017) 797 [arXiv:1612.03631] [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Dimitrios Katsinis
    • 1
    • 2
  • Ioannis Mitsoulas
    • 2
    • 1
  • Georgios Pastras
    • 2
    Email author
  1. 1.Department of Physics, School of Applied Mathematics and Physical SciencesNational Technical UniversityAthensGreece
  2. 2.NCSR “Demokritos”, Institute of Nuclear and Particle PhysicsAghia ParaskeviGreece

Personalised recommendations