Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Dynamical stability of the train of smooth solitary waves to the generalized two-component Camassa-Holm system


Authors: Ting Luo and Min Zhu
Journal: Quart. Appl. Math. 75 (2017), 201-230
MSC (2010): Primary 35B35, 35G25
DOI: https://doi.org/10.1090/qam/1453
Published electronically: July 29, 2016
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Abstract: The present study is concerned with the stability of solitary waves for the generalized two-component Camassa-Holm system derived formally as a model in the shallow-water waves. Using the property of almost monotonicity and the local coercivity of the solitary-wave solution, it is shown that the train of $ N$-smooth solitary waves of this system is dynamically stable to perturbations in energy space with a range of parameters.


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  • [1] T. B. Benjamin, The stability of solitary waves, Proc. Roy. Soc. (London) Ser. A 328 (1972), 153-183. MR 0338584
  • [2] T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A 272 (1972), no. 1220, 47-78. MR 0427868
  • [3] J. L. Bona, W. G. Pritchard, and L. R. Scott, Solitary-wave interaction, Phys. Fluids 23 (1980), 438-441.
  • [4] Alberto Bressan and Adrian Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal. 183 (2007), no. 2, 215-239. MR 2278406, https://doi.org/10.1007/s00205-006-0010-z
  • [5] Alberto Bressan and Adrian Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.) 5 (2007), no. 1, 1-27. MR 2288533, https://doi.org/10.1142/S0219530507000857
  • [6] 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, https://doi.org/10.1103/PhysRevLett.71.1661
  • [7] R. Camassa, D. D. Holm, and J. M. Hyman, A new integral shallow water equation, Adv. Appl. Mech. 31 (1994), 1-33.
  • [8] Robin Ming Chen and Yue Liu, Wave breaking and global existence for a generalized two-component Camassa-Holm system, Int. Math. Res. Not. IMRN 6 (2011), 1381-1416. MR 2806508, https://doi.org/10.1093/imrn/rnq118
  • [9] Robin Ming Chen, Yue Liu, and Zhijun Qiao, Stability of solitary waves and global existence of a generalized two-component Camassa-Holm system, Comm. Partial Differential Equations 36 (2011), no. 12, 2162-2188. MR 2852073, https://doi.org/10.1080/03605302.2011.556695
  • [10] Adrian Constantin, On the inverse spectral problem for the Camassa-Holm equation, J. Funct. Anal. 155 (1998), no. 2, 352-363. MR 1624553, https://doi.org/10.1006/jfan.1997.3231
  • [11] 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
  • [12] Adrian Constantin, Vladimir S. Gerdjikov, and Rossen I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems 22 (2006), no. 6, 2197-2207. MR 2277537, https://doi.org/10.1088/0266-5611/22/6/017
  • [13] Adrian Constantin and Rossen I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A 372 (2008), no. 48, 7129-7132. MR 2474608, https://doi.org/10.1016/j.physleta.2008.10.050
  • [14] 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, https://doi.org/10.1007/s00205-008-0128-2
  • [15] A. Constantin and H. P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math. 52 (1999), no. 8, 949-982. MR 1686969, https://doi.org/10.1002/(SICI)1097-0312(199908)52:8$ \langle $949::AID-CPA3$ \rangle $3.0.CO;2-D
  • [16] Adrian Constantin and Walter A. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000), no. 5, 603-610. MR 1737505, https://doi.org/10.1002/(SICI)1097-0312(200005)53:5$ \langle $603::AID-CPA3$ \rangle $3.3.CO;2-C
  • [17] A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci. 12 (2002), no. 4, 415-422. MR 1915943, https://doi.org/10.1007/s00332-002-0517-x
  • [18] 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, https://doi.org/10.1007/BF01170373
  • [19] Khaled El Dika and Luc Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009), no. 4, 1517-1532. MR 2542735, https://doi.org/10.1016/j.anihpc.2009.02.002
  • [20] Khaled El Dika and Luc Molinet, Stability of multi antipeakon-peakons profile, Discrete Contin. Dyn. Syst. Ser. B 12 (2009), no. 3, 561-577. MR 2525156, https://doi.org/10.3934/dcdsb.2009.12.561
  • [21] Khaled El Dika and Luc Molinet, Exponential decay of $ H^1$-localized solutions and stability of the train of $ N$ solitary waves for the Camassa-Holm equation, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 365 (2007), no. 1858, 2313-2331. MR 2329151, https://doi.org/10.1098/rsta.2007.2011
  • [22] Khaled El Dika and Yvan Martel, Stability of $ N$ solitary waves for the generalized BBM equations, Dyn. Partial Differ. Equ. 1 (2004), no. 4, 401-437. MR 2127579, https://doi.org/10.4310/DPDE.2004.v1.n4.a3
  • [23] Joachim Escher, Olaf Lechtenfeld, and Zhaoyang Yin, Well-posedness and blow-up phenomena for the 2-component Camassa-Holm equation, Discrete Contin. Dyn. Syst. 19 (2007), no. 3, 493-513. MR 2335761, https://doi.org/10.3934/dcds.2007.19.493
  • [24] 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, https://doi.org/10.1016/0167-2789(81)90004-X
  • [25] Guilong Gui and Yue Liu, On the global existence and wave-breaking criteria for the two-component Camassa-Holm system, J. Funct. Anal. 258 (2010), no. 12, 4251-4278. MR 2609545, https://doi.org/10.1016/j.jfa.2010.02.008
  • [26] Guilong Gui and Yue Liu, On the Cauchy problem for the two-component Camassa-Holm system, Math. Z. 268 (2011), no. 1-2, 45-66. MR 2805424, https://doi.org/10.1007/s00209-009-0660-2
  • [27] Katrin Grunert, Helge Holden, and Xavier Raynaud, Global solutions for the two-component Camassa-Holm system, Comm. Partial Differential Equations 37 (2012), no. 12, 2245-2271. MR 3005543, https://doi.org/10.1080/03605302.2012.683505
  • [28] Rossen Ivanov, Two-component integrable systems modelling shallow water waves: the constant vorticity case, Wave Motion 46 (2009), no. 6, 389-396. MR 2598636, https://doi.org/10.1016/j.wavemoti.2009.06.012
  • [29] R. S. Johnson, The Camassa-Holm equation for water waves moving over a shear flow, Fluid Dynam. Res. 33 (2003), no. 1-2, 97-111. In memoriam Prof. Philip Gerald Drazin 1934-2002. MR 1995029, https://doi.org/10.1016/S0169-5983(03)00036-4
  • [30] D. J. Korteweg and G. de Vries, On the change of the form of long waves advancing in rectangular channel, and on a new type of long stationary waves, Phil. Mag. 39 (1895), 422-443.
  • [31] Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490. MR 0235310
  • [32] Jonatan Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations 217 (2005), no. 2, 393-430. MR 2168830, https://doi.org/10.1016/j.jde.2004.09.007
  • [33] 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, https://doi.org/10.1006/jdeq.1999.3683
  • [34] Yvan Martel, Frank Merle, and Tai-Peng Tsai, Stability and asymptotic stability in the energy space of the sum of $ N$ solitons for subcritical gKdV equations, Comm. Math. Phys. 231 (2002), no. 2, 347-373. MR 1946336, https://doi.org/10.1007/s00220-002-0723-2
  • [35] Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412-459. MR 0404890
  • [36] Octavian G. Mustafa, On smooth traveling waves of an integrable two-component Camassa-Holm shallow water system, Wave Motion 46 (2009), no. 6, 397-402. MR 2598637, https://doi.org/10.1016/j.wavemoti.2009.06.011
  • [37] Peter J. Olver, Euler operators and conservation laws of the BBM equation, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 143-160. MR 510408, https://doi.org/10.1017/S0305004100055572
  • [38] 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, https://doi.org/10.1103/PhysRevE.53.1900
  • [39] Jaime Angulo Pava, Nonlinear dispersive equations, Mathematical Surveys and Monographs, vol. 156, American Mathematical Society, Providence, RI, 2009. Existence and stability of solitary and periodic travelling wave solutions. MR 2567568
  • [40] A. B. Shabat and L. Martínez Alonso, On the prolongation of a hierarchy of hydrodynamic chains, New trends in integrability and partial solvability, NATO Sci. Ser. II Math. Phys. Chem., vol. 132, Kluwer Acad. Publ., Dordrecht, 2004, pp. 263-280. MR 2153341, https://doi.org/10.1007/978-94-007-1023-8_10
  • [41] Terence Tao, Why are solitons stable?, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 1-33. MR 2457070, https://doi.org/10.1090/S0273-0979-08-01228-7
  • [42] G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
  • [43] N. J. Zabusky and M. D. Kruskal, Interaction of ``solitons'' in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), 240-243.
  • [44] Pingzheng Zhang and Yue Liu, Stability of solitary waves and wave-breaking phenomena for the two-component Camassa-Holm system, Int. Math. Res. Not. IMRN 11 (2010), 1981-2021. MR 2646352, https://doi.org/10.1093/imrn/rnp211

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Additional Information

Ting Luo
Affiliation: Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019-0408
Email: ting.luo@mavs.uta.edu

Min Zhu
Affiliation: Department of Mathematics, Nanjing Forestry University, Nanjing 310037, People’s Republic of China
Email: zhumin@njfu.edu.cn

DOI: https://doi.org/10.1090/qam/1453
Keywords: Two-component Camassa-Holm system, solitary waves, orbital stability, monotonicity
Received by editor(s): April 14, 2016
Published electronically: July 29, 2016
Article copyright: © Copyright 2016 Brown University

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