Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Katsinis, I. Mitsoulas and G. Pastras, Salient Features of Dressed Elliptic String Solutions on ℝ × S 2, arXiv:1903.01408 [INSPIRE].
M. Spradlin and A. Volovich, Dressing the Giant Magnon, JHEP10 (2006) 012 [hep-th/0607009] [INSPIRE].
C. Kalousios, M. Spradlin and A. Volovich, Dressing the giant magnon II, JHEP03 (2007) 020 [hep-th/0611033] [INSPIRE].
D. Katsinis, I. Mitsoulas and G. Pastras, Elliptic string solutions on ℝ × S 2and their pohlmeyer reduction, Eur. Phys. J.C 78 (2018) 977 [arXiv:1805.09301] [INSPIRE].
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].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP04 (2002) 013 [hep-th/0202021] [INSPIRE].
D.M. Hofman and J.M. Maldacena, Giant Magnons, J. Phys.A 39 (2006) 13095 [hep-th/0604135] [INSPIRE].
R. Ishizeki and M. Kruczenski, Single spike solutions for strings on S 2and S 3, Phys. Rev.D 76 (2007) 126006 [arXiv:0705.2429] [INSPIRE].
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].
A.E. Mosaffa and B. Safarzadeh, Dual spikes: New spiky string solutions, JHEP08 (2007) 017 [arXiv:0705.3131] [INSPIRE].
B.-H. Lee and C. Park, Unbounded Multi Magnon and Spike, J. Korean Phys. Soc.57 (2010) 30 [arXiv:0812.2727] [INSPIRE].
M. Kruczenski, J. Russo and A.A. Tseytlin, Spiky strings and giant magnons on S 5, JHEP10 (2006) 002 [hep-th/0607044] [INSPIRE].
K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys.46 (1976) 207 [INSPIRE].
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. JETP47 (1978) 1017 [INSPIRE].
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].
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].
T.J. Hollowood and J.L. Miramontes, Magnons, their Solitonic Avatars and the Pohlmeyer Reduction, JHEP04 (2009) 060 [arXiv:0902.2405] [INSPIRE].
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].
C.K.R.T. Jones, R. Marangell, P.D. Miller, R.G. Plaza, On the Stability Analysis of Periodic Sine-Gordon Traveling Waves, PhysicaD 251 (2013) 63 [arXiv:1210.0659].
I. Bakas and G. Pastras, On elliptic string solutions in AdS 3and dS 3, JHEP07 (2016) 070 [arXiv:1605.03920] [INSPIRE].
G. Pastras, Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics, 2017, arXiv:1706.07371 [INSPIRE].
B. Wang, Stability of Helicoids in Hyperbolic Three-Dimensional Space, arXiv:1502.04764.
B. Wang, Least Area Spherical Catenoids in Hyperbolic Three-Dimensional Space, arXiv:1204.4943.
G. Pastras, Static elliptic minimal surfaces in AdS 4, Eur. Phys. J.C 77 (2017) 797 [arXiv:1612.03631] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1903.01412
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Katsinis, D., Mitsoulas, I. & Pastras, G. Stability analysis of classical string solutions and the dressing method. J. High Energ. Phys. 2019, 106 (2019). https://doi.org/10.1007/JHEP09(2019)106
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2019)106