Abstract
We investigate soliton collisions in a one-parameter family of scalar field theories in 1+1 dimensions which was first discussed by Christ and Lee [1]. The models have a sextic potential with three local minima, and for suitably small values of the parameter their kinks have an internal structure in the form of two weakly-bound subkinks. We show that for these values of the parameter kink collisions are best understood as an independent sequence of collisions of these subkinks, and that a static mode analysis is not enough to explain resonant structures emerging in this model. We also emphasise the role of radiation and oscillon formation in the collision process.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
N.H. Christ and T.D. Lee, Quantum Expansion of Soliton Solutions, Phys. Rev. D 12 (1975) 1606 [INSPIRE].
N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press (2004) [https://doi.org/10.1017/CBO9780511617034] [INSPIRE].
T. Vachaspati, Kinks and Domain Walls: An Introduction to Classical and Quantum Solitons, Cambridge University Press (2006) [https://doi.org/10.1017/CBO9780511535192].
Y.M. Shnir, Topological and Non-Topological Solitons in Scalar Field Theories, Cambridge University Press (2018) [https://doi.org/10.1017/9781108555623].
P.G. Kevrekidis and J. Cuevas-Maraver, A Dynamical Perspective on the ϕ4 Model, Nonlinear Systems and Complexity, Springer International Publishing (2019) [https://doi.org/10.1007/978-3-030-11839-6].
D.K. Campbell, J.F. Schonfeld and C.A. Wingate, Resonance Structure in Kink-Antikink Interactions in ϕ4 Theory, Physica D 9 (1983) 1 [INSPIRE].
R.H. Goodman and R. Haberman, Kink-Antikink Collisions in the ϕ4 Equation: The n-Bounce Resonance and the Separatrix Map, SIAM Journal on Applied Dynamical Systems 4 (2005) 1195.
V.G. Makhankov, Dynamics of Classical Solitons In Nonintegrable Systems, Phys. Rept. 35 (1978) 1 [INSPIRE].
M. Moshir, Soliton-Antisoliton Scattering and Capture in λϕ4 Theory, Nucl. Phys. B 185 (1981) 318 [INSPIRE].
P. Anninos, S. Oliveira and R.A. Matzner, Fractal structure in the scalar λ(ϕ2 – 1)2 theory, Phys. Rev. D 44 (1991) 1147 [INSPIRE].
P. Dorey, K. Mersh, T. Romańczukiewicz and Y. Shnir, Kink-antikink collisions in the ϕ6 model, Phys. Rev. Lett. 107 (2011) 091602 [arXiv:1101.5951] [INSPIRE].
H. Weigel, Kink-Antikink Scattering in φ4 and ϕ6 Models, J. Phys. Conf. Ser. 482 (2014) 012045 [arXiv:1309.6607] [INSPIRE].
V.A. Gani, A.E. Kudryavtsev and M.A. Lizunova, Kink interactions in the (1 + 1)-dimensional ϕ6 model, Phys. Rev. D 89 (2014) 125009 [arXiv:1402.5903] [INSPIRE].
A. Demirkaya et al., Kink dynamics in a parametric ϕ6 system: a model with controllably many internal modes, JHEP 12 (2017) 071 [arXiv:1706.01193] [INSPIRE].
A. Khare, I.C. Christov and A. Saxena, Successive phase transitions and kink solutions in ϕ8, ϕ10, and ϕ12 field theories, Phys. Rev. E 90 (2014) 023208 [arXiv:1402.6766] [INSPIRE].
T. Romańczukiewicz and Y. Shnir, Some Recent Developments on Kink Collisions and Related Topics, arXiv:1809.04896 [INSPIRE].
D. Bazeia, J.G.F. Campos and A. Mohammadi, Kink-antikink collisions in the ϕ8 model: short-range to long-range journey, JHEP 05 (2023) 116 [arXiv:2303.12482] [INSPIRE].
T. Sugiyama, Kink-antikink collisions in the two-dimensional ϕ4 model, Prog. Theor. Phys. 61 (1979) 1550 [INSPIRE].
P. Dorey and T. Romańczukiewicz, Resonant kink-antikink scattering through quasinormal modes, Phys. Lett. B 779 (2018) 117 [arXiv:1712.10235] [INSPIRE].
J.G.F. Campos and A. Mohammadi, Quasinormal modes in kink excitations and kink-antikink interactions: a toy model, Eur. Phys. J. C 80 (2020) 352 [arXiv:1905.00835] [INSPIRE].
Y.S. Kivshar, Z. Fei and L. Vázquez, Resonant soliton-impurity interactions, Phys. Rev. Lett. 67 (1991) 1177 [INSPIRE].
F. Zhang, Y.S. Kivshar, B.A. Malomed and L. Vázquez, Kink capture by a local impurity in the sine-Gordon model, Phys. Lett. A 159 (1991) 318.
R. Arthur, P. Dorey and R. Parini, Breaking integrability at the boundary: the sine-Gordon model with Robin boundary conditions, J. Phys. A 49 (2016) 165205 [arXiv:1509.08448] [INSPIRE].
P. Dorey et al., Boundary scattering in the ϕ4 model, JHEP 05 (2017) 107 [arXiv:1508.02329] [INSPIRE].
F.C. Lima, F.C. Simas, K.Z. Nobrega and A.R. Gomes, Boundary scattering in the ϕ6 model, JHEP 10 (2019) 147 [arXiv:1808.06703] [INSPIRE].
P. Dorey et al., Resonance structures in kink-antikink collisions in a deformed sine-Gordon model, JHEP 09 (2021) 145 [arXiv:2106.09560] [INSPIRE].
A. Alonso Izquierdo, L.M. Nieto and J. Queiroga-Nunes, Scattering between wobbling kinks, Phys. Rev. D 103 (2021) 045003 [arXiv:2007.15517] [INSPIRE].
P. Dorey, T. Romańczukiewicz and Y. Shnir, Staccato radiation from the decay of large amplitude oscillons, Phys. Lett. B 806 (2020) 135497 [arXiv:1910.04128] [INSPIRE].
F.C. Simas, A.R. Gomes, K.Z. Nobrega and J.C.R.E. Oliveira, Suppression of two-bounce windows in kink-antikink collisions, JHEP 09 (2016) 104 [arXiv:1605.05344] [INSPIRE].
J. Ashcroft et al., Head butting sheep: Kink Collisions in the Presence of False Vacua, J. Phys. A 49 (2016) 365203 [arXiv:1604.08413] [INSPIRE].
A.R. Gomes, F.C. Simas, K.Z. Nobrega and P.P. Avelino, False vacuum decay in kink scattering, JHEP 10 (2018) 192 [arXiv:1805.00991] [INSPIRE].
C. Adam, K. Oles, T. Romańczukiewicz and A. Wereszczynski, Kink-antikink collisions in a weakly interacting ϕ4 model, Phys. Rev. E 102 (2020) 062214 [arXiv:1912.09371] [INSPIRE].
C. Adam, K. Oles, T. Romańczukiewicz and A. Wereszczynski, Kink-antikink scattering in the ϕ4 model without static intersoliton forces, Phys. Rev. D 101 (2020) 105021 [arXiv:1909.06901] [INSPIRE].
M. Sanati and A. Saxena, Half-kink lattice solution of the ϕ6 model, J. Phys. A 32 (1999) 4311 [INSPIRE].
T.S. Mendonça and H.P. de Oliveira, A note about a new class of two-kinks, JHEP 06 (2015) 133 [arXiv:1504.07315] [INSPIRE].
Y. Zhong et al., Collision of two kinks with inner structure, JHEP 02 (2020) 153 [arXiv:1906.02920] [INSPIRE].
P. Forgács, Á. Lukács and T. Romańczukiewicz, Plane waves as tractor beams, Phys. Rev. D 88 (2013) 125007 [arXiv:1303.3237] [INSPIRE].
P. Forgács, Á. Lukács and T. Romańczukiewicz, Negative radiation pressure exerted on kinks, Phys. Rev. D 77 (2008) 125012 [arXiv:0802.0080] [INSPIRE].
T. Romańczukiewicz, Interaction between kink and radiation in ϕ4 model, Acta Phys. Polon. B 35 (2004) 523 [hep-th/0303058] [INSPIRE].
N.S. Manton, Topology in the Weinberg-Salam Theory, Phys. Rev. D 28 (1983) 2019 [INSPIRE].
F.R. Klinkhamer and N.S. Manton, A Saddle Point Solution in the Weinberg-Salam Theory, Phys. Rev. D 30 (1984) 2212 [INSPIRE].
I. Takyi and H. Weigel, Collective Coordinates in One-Dimensional Soliton Models Revisited, Phys. Rev. D 94 (2016) 085008 [arXiv:1609.06833] [INSPIRE].
C. Adam et al., Multikink scattering in the ϕ6 model revisited, Phys. Rev. D 106 (2022) 125003 [arXiv:2209.08849] [INSPIRE].
T. Romańczukiewicz, Could the primordial radiation be responsible for vanishing of topological defects?, Phys. Lett. B 773 (2017) 295 [arXiv:1706.05192] [INSPIRE].
T. Tao, Why are solitons stable?, Bull. Am. Math. Soc. 46 (2009) 1 [arXiv:0802.2408].
P. Bizoń, B. Cownden and M. Maliborski, Characteristic approach to the soliton resolution, Nonlinearity 35 (2022) 4585 [arXiv:2112.11249] [INSPIRE].
D.K. Campbell, M. Peyrard and P. Sodano, Kink-antikink interactions in the double Sine-Gordon equation, Physica D 19 (1986) 165 [INSPIRE].
J.G.F. Campos and A. Mohammadi, Wobbling double sine-Gordon kinks, JHEP 09 (2021) 067 [arXiv:2103.04908] [INSPIRE].
V.A. Gani et al., Scattering of the double sine-Gordon kinks, Eur. Phys. J. C 78 (2018) 345 [arXiv:1711.01918] [INSPIRE].
E. Forest and R.D. Ruth, Fourth order symplectic integration, Physica D 43 (1990) 105 [INSPIRE].
H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A 150 (1990) 262 [INSPIRE].
Acknowledgments
We would like to thank Nick Manton for interesting conversations and many insightful remarks. The research of PED was supported in part by the National Science Foundation under Grant No. NSF PHY-1748958, and in part from the STFC under consolidated grant ST/T000708/1; he also wishes to thanks the African Institute for Mathematical Sciences South Africa for hospitality while this work was completed. TR wishes to thank National Science Centre, grant number 2019/35/B/ST2/00059 and the Priority Research Area under the program Excellence Initiative — Research University at the Jagiellonian University in Kraków. YS gratefully acknowledges the support of the Alexander von Humboldt Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2304.11710
Rights and permissions
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.
About this article
Cite this article
Dorey, P., Gorina, A., Romańczukiewicz, T. et al. Collisions of weakly-bound kinks in the Christ-Lee model. J. High Energ. Phys. 2023, 45 (2023). https://doi.org/10.1007/JHEP09(2023)045
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)045