Orbital stability of smooth solitary waves for the Degasperis-Procesi equation
HTML articles powered by AMS MathViewer
- by Ji Li, Yue Liu and Qiliang Wu
- Proc. Amer. Math. Soc. 151 (2023), 151-160
- DOI: https://doi.org/10.1090/proc/16087
- Published electronically: September 2, 2022
- HTML | PDF | Request permission
Abstract:
The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of $L^3$, but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the $L^2$-norm, resulting in $L^3$ higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of $L^2$ norm for DP equation, which cannot be used to control the $L^3$ higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the $L^\infty$ orbital norm of the perturbation is bounded above by a function of its $L^2$ orbital norm, yielding the higher order control and the orbital stability in the $L^2\cap L^\infty$ space.References
- 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
- 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>3.3.CO;2-C
- A. Constantin and W. A. Strauss, Stability of the Camassa-Holm solitons, J. Nonlinear Sci. 12 (2002), no. 4, 415–422. MR 1915943, DOI 10.1007/s00332-002-0517-x
- A. Degasperis and M. Procesi, Asymptotic integrability, Symmetry and perturbation theory (Rome, 1998) World Sci. Publ., River Edge, NJ, 1999, pp. 23–37. MR 1844104
- Joachim Escher, Yue Liu, and Zhaoyang Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal. 241 (2006), no. 2, 457–485. MR 2271927, DOI 10.1016/j.jfa.2006.03.022
- Gadi Fibich, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 192, Springer, Cham, 2015. Singular solutions and optical collapse. MR 3308230, DOI 10.1007/978-3-319-12748-4
- 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
- Manoussos Grillakis, Jalal Shatah, and Walter Strauss, Stability theory of solitary waves in the presence of symmetry. I, J. Funct. Anal. 74 (1987), no. 1, 160–197. MR 901236, DOI 10.1016/0022-1236(87)90044-9
- Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin-New York, 1975, pp. 25–70. MR 407477
- D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Philos. Mag. (5) 39 (1895), no. 240, 422–443. MR 3363408, DOI 10.1080/14786449508620739
- Bashar Khorbatly and Luc Molinet, On the orbital stability of the Degasperis-Procesi antipeakon-peakon profile, J. Differential Equations 269 (2020), no. 6, 4799–4852. MR 4104459, DOI 10.1016/j.jde.2020.03.045
- Todd Kapitula and Keith Promislow, Spectral and dynamical stability of nonlinear waves, Applied Mathematical Sciences, vol. 185, Springer, New York, 2013. With a foreword by Christopher K. R. T. Jones. MR 3100266, DOI 10.1007/978-1-4614-6995-7
- Ji Li, Yue Liu, and Qiliang Wu, Spectral stability of smooth solitary waves for the Degasperis-Procesi equation, J. Math. Pures Appl. (9) 142 (2020), 298–314 (English, with English and French summaries). MR 4149693, DOI 10.1016/j.matpur.2020.08.003
- 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
- Zhiwu Lin and Chongchun Zeng, Instability, index theorem, and exponential trichotomy for linear Hamiltonian PDEs, Mem. Amer. Math. Soc. 275 (2022), no. 1347, v+136. MR 4352468, DOI 10.1090/memo/1347
- Yue Liu and Zhaoyang Yin, Global existence and blow-up phenomena for the Degasperis-Procesi equation, Comm. Math. Phys. 267 (2006), no. 3, 801–820. MR 2249792, DOI 10.1007/s00220-006-0082-5
- Catherine Sulem and Pierre-Louis Sulem, The nonlinear Schrödinger equation, Applied Mathematical Sciences, vol. 139, Springer-Verlag, New York, 1999. Self-focusing and wave collapse. MR 1696311
- Michael I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472–491. MR 783974, DOI 10.1137/0516034
- Michael I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), no. 1, 51–67. MR 820338, DOI 10.1002/cpa.3160390103
- Zhaoyang Yin, On the Cauchy problem for an integrable equation with peakon solutions, Illinois J. Math. 47 (2003), no. 3, 649–666. MR 2007229
- Zhaoyang Yin, Global solutions to a new integrable equation with peakons, Indiana Univ. Math. J. 53 (2004), no. 4, 1189–1209. MR 2095454, DOI 10.1512/iumj.2004.53.2479
Bibliographic Information
- Ji Li
- Affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, People’s Republic of China
- Email: liji@hust.edu.cn
- Yue Liu
- Affiliation: Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019-0408
- Email: yliu@uta.edu
- Qiliang Wu
- Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
- MR Author ID: 1059244
- Email: wuq@ohio.edu
- Received by editor(s): December 7, 2021
- Received by editor(s) in revised form: March 8, 2022, and March 11, 2022
- Published electronically: September 2, 2022
- Additional Notes: The work of the first author was partially supported by the NSFC grant 11771161, 12171174. The work of the second author was partially supported by the Simons Foundation grant 499875. The work of the third author was partially supported by the NSF grant DMS-1815079
The third author is the corresponding author. - Communicated by: Catherine Sulem
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 151-160
- MSC (2020): Primary 35Q35, 35Q51
- DOI: https://doi.org/10.1090/proc/16087
- MathSciNet review: 4504615