Zhijun Qiao, Xianqi Li, “An integrable equation with nonsmooth solitons”, TMF, 167:2 (2011), 214–221; Theoret. and Math. Phys., 167:2 (2011), 584

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Teoreticheskaya i Matematicheskaya Fizika, 2011, Volume 167, Number 2, Pages 214–221
DOI: https://doi.org/10.4213/tmf6635 (Mi tmf6635)

This article is cited in 51 scientific papers (total in 51 papers)

An integrable equation with nonsmooth solitons

Zhijun Qiao, Xianqi Li

Department of Mathematics, The University of Texas-Pan American, Edinburg, USA

Full-text PDF (407 kB) Citations (51)

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DOI: https://doi.org/10.4213/tmf6635

Abstract: We present the bi-Hamiltonian structure and Lax pair of the equation

ρ_{t} = b u_{x} + (1 / 2) [(u^{2} - u_{x}^{2}) ρ]_{x}

$\rho_t= bu_x+(1/2)[(u^2-u_x^2)\rho]_x$ , where

ρ = u - u_{x x}

$\rho=u-u_{xx}$ and

b = c o n s t

$b=\mathrm{const}$ , which guarantees its integrability in the Lax pair sense. We study nonsmooth soliton solutions of this equation and show that under the vanishing boundary condition

u \to 0

$u\to0$ at the space and time infinities, the equation has both “W/M-shape” peaked soliton (peakon) and cusped soliton (cuspon) solutions.

Keywords: integrable equation, Lax pair, peakon, cuspon.

Received: 03.06.2009
Revised: 26.10.2010

English version:
Theoretical and Mathematical Physics, 2011, Volume 167, Issue 2, Pages 584–589
DOI: https://doi.org/10.1007/s11232-011-0044-8

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: Zhijun Qiao, Xianqi Li, “An integrable equation with nonsmooth solitons”, TMF, 167:2 (2011), 214–221; Theoret. and Math. Phys., 167:2 (2011), 584–589

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/tmf6635

https://doi.org/10.4213/tmf6635

https://www.mathnet.ru/eng/tmf/v167/i2/p214

Erratum

Erratum in Zhijun Qiao and Xianqi Li (Vol. 167, No. 2, May, 2011, pp. 584–589)
TMF, 2011, 169:1, 176

This publication is cited in the following 51 articles:

Xingjie Yan, Ruofan An, Yu Zhang, Xingxing Liu, “Orbital stability of the periodic peakons for the higher-order modified Camassa-Holm equation”, DCDS-S, 2024
Dandan He, Tongjie Deng, Kelei Zhang, “Orbital stability of peakons and multi-peakons for a generalized quintic-septic Camassa-Holm type equation”, Quart. Appl. Math., 2024
Bao-Feng Feng, Heng-Chun Hu, Han-Han Sheng, Wei Yin, Guo-Fu Yu, “Integrable Semi-Discretization for a Modified Camassa–Holm Equation with Cubic Nonlinearity”, SIGMA, 20 (2024), 091, 14 pp.
Wen-Yu Zhou, Shou-Fu Tian, Zhi-Qiang Li, “On Riemann–Hilbert problem and multiple high-order pole solutions to the cubic Camassa–Holm equation”, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2024, 1
Weifang Weng, Zhijun Qiao, Zhenya Yan, “Wave-breaking analysis and weak multi-peakon solutions for a generalized cubic–quintic Camassa–Holm type equation”, Monatsh Math, 200:3 (2023), 667
Gezi Chong, Ying Fu, Hao Wang, “Orbital stability of periodic peakons for the higher-order modified Camassa–Holm equation”, Monatsh Math, 2023
Ke Jiang, Feng Cao, “Lyapunov-type stability criterion for periodic generalized Camassa–Holm equations”, Nonlinear Analysis: Real World Applications, 74 (2023), 103940
Qin G., Yan Zh., Guo B., “The Cauchy Problem and Wave-Breaking Phenomenon For a Generalized Sine-Type Forq/Mch Equation”, Mon.heft. Math., 198:3 (2022), 619–640
Qin G., Yan Zh., Guo B., “A Sine-Type Camassa-Holm Equation: Local Well-Posedness, Holder Continuity, and Wave-Breaking Analysis”, Mon.heft. Math., 2022
Qin G., Yan Zh., Guo B., “The Cauchy Problem and Multi-Peakons For the Mch-Novikov-Ch Equation With Quadratic and Cubic Nonlinearities”, J. Dyn. Differ. Equ., 2022
Li J., “Orbital Stability of Peakons For the Modified Camassa-Holm Equation”, Acta. Math. Sin.-English Ser., 38:1 (2022), 148–160
Han‐Han Sheng, Guo‐Fu Yu, Bao‐Feng Feng, “An integrable semidiscretization of the modified Camassa–Holm equation with linear dispersion term”, Stud Appl Math, 149:1 (2022), 230
Haotian Wang, Qin Zhou, Wenjun Liu, “Exact analysis and elastic interaction of multi-soliton for a two-dimensional Gross-Pitaevskii equation in the Bose-Einstein condensation”, Journal of Advanced Research, 38 (2022), 179
Aiyong Chen, Hao Yu, Xiaokai He, “2-Peakon Solutions and Non-uniqueness of the Fokas–Olver–Rosenau–Qiao Equation with Higher-Order Nonlinearity”, Bull. Malays. Math. Sci. Soc., 45:6 (2022), 3401
Guoquan Qin, Zhenya Yan, Boling Guo, “Orbital stability of peakon solutions for a generalized higher-order Camassa–Holm equation”, Z. Angew. Math. Phys., 73:3 (2022)
Yiling Yang, Engui Fan, “On the long-time asymptotics of the modified Camassa-Holm equation in space-time solitonic regions”, Advances in Mathematics, 402 (2022), 108340
Zhang L., Qiao Zh., “Global-in-Time Solvability and Blow-Up For a Non-Isospectral Two-Component Cubic Camassa-Holm System in a Critical Besov Space”, J. Differ. Equ., 274 (2021), 414–460
Moon B., “Orbital Stability of Periodic Peakons For the Generalized Modified Camassa-Holm Equation”, Discret. Contin. Dyn. Syst.-Ser. S, 14:12 (2021), 4409–4437
Anco S.C. Wang B., “A Formula For Symmetry Recursion Operators From Non-Variational Symmetries of Partial Differential Equations”, Lett. Math. Phys., 111:3 (2021), 70
Devi A.D., Krishnakumar K., Sinuvasan R., Leach P.G.L., “Symmetries and Integrability of the Modified Camassa-Holm Equation With An Arbitrary Parameter”, Pramana-J. Phys., 95:2 (2021), 85

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

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