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
MathSciNet review:
3614495
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Additional Information
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.
References
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References
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- 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.
- Peter D. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467–490. MR 0235310
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- Robert M. Miura, The Korteweg-de Vries equation: a survey of results, SIAM Rev. 18 (1976), no. 3, 412–459. MR 0404890
- 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, DOI https://doi.org/10.1016/j.wavemoti.2009.06.011
- 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, DOI https://doi.org/10.1017/S0305004100055572
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- 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
- 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, DOI https://doi.org/10.1007/978-94-007-1023-8_10
- Terence Tao, Why are solitons stable?, Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 1–33. MR 2457070, DOI https://doi.org/10.1090/S0273-0979-08-01228-7
- G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. MR 0483954
- 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.
- 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, DOI 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
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