V. B. Matveev, “Positons: Slowly Decreasing Analogues of Solitons”, TMF, 131:1 (2002), 44–61; Theoret. and Math. Phys., 131:1 (2002), 483

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Teoreticheskaya i Matematicheskaya Fizika, 2002, Volume 131, Number 1, Pages 44–61
DOI: https://doi.org/10.4213/tmf1946 (Mi tmf1946)

This article is cited in 86 scientific papers (total in 86 papers)

Positons: Slowly Decreasing Analogues of Solitons

V. B. Matveev^abc

^a St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
^b Max Planck Institute for Mathematics
^c Université de Bourgogne

Full-text PDF (274 kB) Citations (86)

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DOI: https://doi.org/10.4213/tmf1946

Abstract: We present an introduction to positon theory, almost never covered in the Russian scientific literature. Positons are long-range analogues of solitons and are slowly decreasing, oscillating solutions of nonlinear integrable equations of the KdV type. Positon and soliton-positon solutions of the KdV equation, first constructed and analyzed about a decade ago, were then constructed for several other models: for the mKdV equation, the Toda chain, the NS equation, as well as the sinh-Gordon equation and its lattice analogue. Under a proper choice of the scattering data, the one-positon and multipositon potentials have a remarkable property: the corresponding reflection coefficient is zero, but the transmission coefficient is unity (as is known, the latter does not hold for the standard short-range reflectionless potentials).

English version:
Theoretical and Mathematical Physics, 2002, Volume 131, Issue 1, Pages 483–497
DOI: https://doi.org/10.1023/A:1015149618529

Bibliographic databases:

Language: Russian

Citation: V. B. Matveev, “Positons: Slowly Decreasing Analogues of Solitons”, TMF, 131:1 (2002), 44–61; Theoret. and Math. Phys., 131:1 (2002), 483–497

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf1946

https://doi.org/10.4213/tmf1946

https://www.mathnet.ru/eng/tmf/v131/i1/p44

Erratum

Positons: Slowly Decreasing Analogues of Solitons [Erratum]
V. B. Matveev
TMF, 2002, 131:2, 352

This publication is cited in the following 86 articles:

Shuzhi Liu, Deqin Qiu, “The exploding solitons of the sine–Gordon equation”, Applied Mathematics Letters, 160 (2025), 109314
Jiaqing Shan, Maohua Li, “The breather, breather-positon, rogue wave for the reverse space–time nonlocal short pulse equation in nonzero background”, Wave Motion, 133 (2025), 103448
Ping Li, Jingsong He, Maohua Li, “The higher-order positon and breather-positon solutions for the complex short pulse equation”, Nonlinear Dyn, 112:12 (2024), 10239
Shuzhi Liu, Deqin Qiu, “WITHDRAWN: Dynamics of solitons for the sine–Gordon equation”, Results in Physics, 2024, 107798
Tao Xu, Jinyan Zhu, “Soliton molecules and breather positon solutions for the coupled modified nonlinear Schrödinger equation”, Wave Motion, 129 (2024), 103347
K. Thulasidharan, N. Vishnu Priya, S. Monisha, M. Senthilvelan, “Predicting positon solutions of a family of nonlinear Schrödinger equations through deep learning algorithm”, Physics Letters A, 511 (2024), 129551
S. Monisha, M. Senthilvelan, Springer Proceedings in Physics, 405, Proceedings of the 2nd International Conference on Nonlinear Dynamics and Applications (ICNDA 2024), Volume 1, 2024, 139
Jiaqing Shan, Maohua Li, “The dynamic of the positons for the reverse space–time nonlocal short pulse equation”, Physica D: Nonlinear Phenomena, 470 (2024), 134419
Runjia LUO, Guoquan ZHOU, “Double-Pole Solution and Soliton-Antisoliton Pair Solution of MNLSE/DNLSE Based upon Hirota Method”, Wuhan Univ. J. Nat. Sci., 29:5 (2024), 430
S. Monisha, N. Vishnu Priya, M. Senthilvelan, “Degenerate soliton solutions and their interactions in coupled Hirota equation with trivial and nontrivial background”, Nonlinear Dyn, 111:23 (2023), 21877
Kannan Manikandan, Nurzhan Serikbayev, Shunmuganathan P. Vijayasree, Devarasu Aravinthan, “Controlling Matter-Wave Smooth Positons in Bose–Einstein Condensates”, Symmetry, 15:8 (2023), 1585
Alexei Rybkin, “Norming Constants of Embedded Bound States and Bounded Positon Solutions of the Korteweg-de Vries Equation”, Commun. Math. Phys., 401:3 (2023), 2433
K. Manikandan, N. Serikbayev, M. Manigandan, M. Sabareeshwaran, “Dynamical evolutions of optical smooth positons in variable coefficient nonlinear Schrödinger equation with external potentials”, Optik, 288 (2023), 171203
Kazuyuki Yagasaki, “Integrability of the Zakharov-Shabat Systems by Quadrature”, Commun. Math. Phys., 400:1 (2023), 315
Feng Yuan, Behzad Ghanbari, “Positon and hybrid solutions for the (2+1)-dimensional complex modified Korteweg–de Vries equations”, Chinese Phys. B, 32:4 (2023), 040201
Santanu Raut, Wen-Xiu Ma, Ranjan Barman, Subrata Roy, “A non-autonomous Gardner equation and its integrability: Solitons, positons and breathers”, Chaos, Solitons & Fractals, 176 (2023), 114089
Deqin Qiu, Yongshuai Zhang, “Classification of solutions of the generalized mixed nonlinear Schrödinger equation”, Theoret. and Math. Phys., 211:3 (2022), 838–855
Sergei Grudsky, Alexei Rybkin, “The inverse scattering transform for weak Wigner–von Neumann type potentials *”, Nonlinearity, 35:5 (2022), 2175
Efim Pelinovsky, Tatiana Talipova, Ekaterina Didenkulova, “Rational Solitons in the Gardner-Like Models”, Fluids, 7:9 (2022), 294
N. Vishnu Priya, S. Monisha, M. Senthilvelan, Govindan Rangarajan, “Nth-order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation”, Eur. Phys. J. Plus, 137:5 (2022)

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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