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Online ISSN 1552-4485; Print ISSN 0033-569X
Orbital stability of peakons and multi-peakons for a generalized quintic-septic Camassa-Holm type equation
Authors:
Dandan He, Tongjie Deng and Kelei Zhang
Journal:
Quart. Appl. Math. 83 (2025), 59-96
MSC (2020):
Primary 37K45; Secondary 35G20, 35D30
DOI:
https://doi.org/10.1090/qam/1699
Published electronically:
June 21, 2024
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: In this paper, we consider the generalized quintic-septic Camassa-Holm (gqsCH) equation, which is actually an extension of the quintic CH equation and the septic CH equation. We first prove the existence of the single peakon. Then, by constructing certain Lyapunov functionals, we prove the stability of peakons in the energy space $H^1(\mathbb {R})$-norm. Finally, we also prove that the sum of $N$ sufficiently decoupled peakons is orbitally stable in the energy space by using energy argument, combining the method of the orbital stability of single peakons with monotonicity of the local energy norm.
- S. Anco and E. Recio, A general family of multi-peakon equations and their properties. J. Phys A: Math. Theor. 52 (2019), 125–203.
- Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661–1664. MR 1234453, DOI 10.1103/PhysRevLett.71.1661
- Jean-Yves Chemin, Localization in Fourier space and Navier-Stokes system, Phase space analysis of partial differential equations. Vol. I, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004, pp. 53–135. MR 2144406
- Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165–186. MR 2481064, DOI 10.1007/s00205-008-0128-2
- Adrian Constantin, Existence of permanent and breaking waves for a shallow water equation: a geometric approach, Ann. Inst. Fourier (Grenoble) 50 (2000), no. 2, 321–362 (English, with English and French summaries). MR 1775353, DOI 10.5802/aif.1757
- Adrian Constantin and Joachim Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math. 181 (1998), no. 2, 229–243. MR 1668586, DOI 10.1007/BF02392586
- Adrian Constantin and Joachim Escher, On the blow-up rate and the blow-up set of breaking waves for a shallow water equation, Math. Z. 233 (2000), no. 1, 75–91. MR 1738352, DOI 10.1007/PL00004793
- Adrian Constantin and Joachim Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 2, 303–328. MR 1631589
- Adrian Constantin and Walter A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000), no. 5, 603–610. MR 1737505, DOI 10.1002/(sici)1097-0312(200005)53:5<603::aid-cpa3>3.0.co;2-l
- Adrian Constantin and Luc Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D 157 (2001), no. 1-2, 75–89. MR 1854962, DOI 10.1016/S0167-2789(01)00298-6
- H.-H. Dai, Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod, Acta Mech. 127 (1998), no. 1-4, 193–207. MR 1606738, DOI 10.1007/BF01170373
- Raphaël Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations 14 (2001), no. 8, 953–988. MR 1827098
- Tongjie Deng and Aiyong Chen, Orbital stability of the sum of $N$ peakons for the generalized modified Camassa-Holm equation, Monatsh. Math. 202 (2023), no. 2, 229–262. MR 4641683, DOI 10.1007/s00605-022-01766-2
- Khaled El Dika and Luc Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 4, 1517–1532. MR 2542735, DOI 10.1016/j.anihpc.2009.02.002
- B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47–66. MR 636470, DOI 10.1016/0167-2789(81)90004-X
- A. S. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math. 39 (1995), no. 1-3, 295–305. KdV ’95 (Amsterdam, 1995). MR 1329566, DOI 10.1007/BF00994638
- B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47–66. MR 636470, DOI 10.1016/0167-2789(81)90004-X
- Benno Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D 95 (1996), no. 3-4, 229–243. MR 1406283, DOI 10.1016/0167-2789(96)00048-6
- Ying Fu, Guilong Gui, Yue Liu, and Changzheng Qu, On the Cauchy problem for the integrable modified Camassa-Holm equation with cubic nonlinearity, J. Differential Equations 255 (2013), no. 7, 1905–1938. MR 3072676, DOI 10.1016/j.jde.2013.05.024
- Zihua Guo, Xiaochuan Liu, Xingxing Liu, and Changzheng Qu, Stability of peakons for the generalized modified Camassa-Holm equation, J. Differential Equations 266 (2019), no. 12, 7749–7779. MR 3944240, DOI 10.1016/j.jde.2018.12.014
- A. Alexandrou Himonas and Dionyssios Mantzavinos, Hölder continuity for the Fokas-Olver-Rosenau-Qiao equation, J. Nonlinear Sci. 24 (2014), no. 6, 1105–1124. MR 3275219, DOI 10.1007/s00332-014-9212-y
- Yi A. Li and Peter J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (2000), no. 1, 27–63. MR 1741872, DOI 10.1006/jdeq.1999.3683
- Xingxing Liu, Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity, Discrete Contin. Dyn. Syst. 38 (2018), no. 11, 5505–5521. MR 3917778, DOI 10.3934/dcds.2018242
- Xingxing Liu, Stability in the energy space of the sum of $N$ peakons for a modified Camassa-Holm equation with higher-order nonlinearity, J. Math. Phys. 59 (2018), no. 12, 121505, 19. MR 3885135, DOI 10.1063/1.5034143
- Xiaochuan Liu, Yue Liu, and Changzheng Qu, Orbital stability of the train of peakons for an integrable modified Camassa-Holm equation, Adv. Math. 255 (2014), 1–37. MR 3167476, DOI 10.1016/j.aim.2013.12.032
- Zhiwu Lin and Yue Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62 (2009), no. 1, 125–146. MR 2460268, DOI 10.1002/cpa.20239
- Byungsoo Moon, Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation, Discrete Contin. Dyn. Syst. Ser. S 14 (2021), no. 12, 4409–4437. MR 4347355, DOI 10.3934/dcdss.2021123
- Peter J. Olver and Philip Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3) 53 (1996), no. 2, 1900–1906. MR 1401317, DOI 10.1103/PhysRevE.53.1900
- Zhijun Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (2006), no. 11, 112701, 9. MR 2278659, DOI 10.1063/1.2365758
- Zhijun Qiao and Xianqi Li, An integrable equation with nonsmooth solitons, Theoret. and Math. Phys. 167 (2011), no. 2, 584–589. Russian version appears in Teoret. Mat. Fiz. 167 (2011), no. 2, 214–221. MR 3166366, DOI 10.1007/s11232-011-0044-8
- Changzheng Qu, Xiaochuan Liu, and Yue Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonlinearity, Comm. Math. Phys. 322 (2013), no. 3, 967–997. MR 3079338, DOI 10.1007/s00220-013-1749-3
- Zhong Wang and Yue Liu, Stability of smooth multi-solitons for the Camassa-Holm equation, Calc. Var. Partial Differential Equations 61 (2022), no. 2, Paper No. 51, 36. MR 4375797, DOI 10.1007/s00526-021-02175-3
- Weifang Weng, Zhijun Qiao, and Zhenya Yan, Wave-breaking analysis and weak multi-peakon solutions for a generalized cubic-quintic Camassa-Holm type equation, Monatsh. Math. 200 (2023), no. 3, 667–713. MR 4555926, DOI 10.1007/s00605-022-01699-w
- Meiling Yang, Yongsheng Li, and Yongye Zhao, On the Cauchy problem of generalized Fokas-Olver-Resenau-Qiao equation, Appl. Anal. 97 (2018), no. 13, 2246–2268. MR 3851153, DOI 10.1080/00036811.2017.1359565
- Shaojie Yang, Blow-up phenomena for the generalized FORQ/MCH equation, Z. Angew. Math. Phys. 71 (2020), no. 1, Paper No. 20, 13. MR 4049257, DOI 10.1007/s00033-019-1241-9
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Additional Information
Dandan He
Affiliation:
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, People’s Republic of China
Email:
h1362740596@163.com
Tongjie Deng
Affiliation:
School of Mathematics and Physics, Center for Applied Mathematics of Guangxi, Guangxi Minzu University, Nanning 530006, People’s Republic of China
MR Author ID:
1503929
Email:
tongjiedeng1@163.com
Kelei Zhang
Affiliation:
Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, People’s Republic of China
ORCID:
0000-0002-9239-1410
Email:
keleizhang@163.com
Keywords:
Generalized quintic-septic CH equation,
peakons,
multi-peakons,
orbital stability
Received by editor(s):
January 26, 2024
Published electronically:
June 21, 2024
Additional Notes:
This work was supported by the Guangxi Key Laboratory of Cryptography and formation Security (No. GCIS202134).
Tongjie Deng is the corresponding author.
Article copyright:
© Copyright 2024
Brown University