Orbital and asymptotic stability of a train of peakons for the Novikov equation

José Manuel Palacios

doi:10.3934/dcds.2020372

Article Contents

2021, Volume 41, Issue 5: 2475-2518. Doi: 10.3934/dcds.2020372

This issue Previous Article Stabilizability in optimization problems with unbounded data Next Article

Orbital and asymptotic stability of a train of peakons for the Novikov equation

José Manuel Palacios

Institut Denis Poisson, Université de Tours, Université d'Orleans, CNRS, Parc Grandmont 37200, Tours, France

Received: April 30, 2020

Revised: August 31, 2020

Early access: November 2020

Published: May 2021

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Abstract

The Novikov equation is an integrable Camassa-Holm type equation with cubic nonlinearity. One of the most important features of this equation is the existence of peakon and multi-peakon solutions, i.e. peaked traveling waves behaving as solitons. This paper aims to prove both, the orbital and asymptotic stability of peakon trains solutions, i.e. multi-peakon solutions such that their initial configuration is increasingly ordered. Furthermore, we give an improvement of the orbital stability of a single peakon so that we can drop the non-negativity hypothesis on the momentum density. The same result also holds for the orbital stability for peakon trains, i.e. in this latter case we can also avoid assuming non-negativity of the initial momentum density. Finally, as a corollary of these results together with some asymptotic formulas for the position and momenta vectors for multi-peakon solutions, we obtain the orbital and asymptotic stability for initially not well-ordered multipeakons.

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Mathematics Subject Classification: 35Q35, 35Q51, 35C08, 35B35, 37K40.

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[1]	B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics, Invent. Math., 171 (2008), 485-541. doi: 10.1007/s00222-007-0088-4.
[2]	A. Bressan and A. Constantin, Global dissipative solutions of the Camassa-Holm equation, Anal. Appl. (Singap.), 5 (2007), 1-27. doi: 10.1142/S0219530507000857.
[3]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.
[4]	R. Camassa and D. D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[5]	R. M. Chen, F. Guo, Y. Liu and C. Qu, Analysis on the blow-up of solutions to a class of integrable peakon equations, J. Funct. Anal., 270 (2016), 2343-2374. doi: 10.1016/j.jfa.2016.01.017.
[6]	A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.
[7]	A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 303-328.
[8]	A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Rat. Mech. Anal., 192 (2009), 165-186. doi: 10.1007/s00205-008-0128-2.
[9]	A. Constantin and L. Molinet, Global weak solutions for a shallow water equation, Commun. Math. Phys., 211 (2000), 45-61. doi: 10.1007/s002200050801.
[10]	A. Constantin and L. Molinet, Orbital stability of solitary waves for a shallow water equation, Phys. D, 157 (2001), 75-89. doi: 10.1016/S0167-2789(01)00298-6.
[11]	A. Constantin and W. A. Strauss, Stability of peakons, Comm. Pure Appl. Math., 53 (2000), 603-610. doi: 10.1002/(SICI)1097-0312(200005)53:5<603::AID-CPA3>3.0.CO;2-L.
[12]	R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.
[13]	A. Degasperis, D. D. Kholm and A. N. I. Khone, A new integrable equation with peakon solution, Theor. Math. Phys., 133 (2002), 1463-1474. doi: 10.1023/A:1021186408422.
[14]	A. Degasperis and M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Sci. Publ., River Edge, NJ, (1999), 23–37.
[15]	K. El Dika and L. 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), 2313-2331. doi: 10.1098/rsta.2007.2011.
[16]	K. El Dika and L. Molinet, Stability of multipeakons, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), 1517-1532. doi: 10.1016/j.anihpc.2009.02.002.
[17]	B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Physica D, 4 (1981/1982), 47-66. doi: 10.1016/0167-2789(81)90004-X.
[18]	A. A. Himonas and C. Holliman, The Cauchy problem for the Novikov equation, Nonlinearity, 25 (2012), 449-479. doi: 10.1088/0951-7715/25/2/449.
[19]	H. Holden and X. Raynaud, Global conservative multipeakon solutions of the Camassa-Holm equation, J. Hyperbolic Differ. Equ., 4 (2007), 39-64. doi: 10.1142/S0219891607001045.
[20]	A. N. W. Hone and J. Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A, 41 (2008), 372002, 10 pp. doi: 10.1088/1751-8113/41/37/372002.
[21]	A. N. W. Hone, H. Lundmark and J. Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ., 6 (2009), 253-289. doi: 10.4310/DPDE.2009.v6.n3.a3.
[22]	R. S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech., 455 (2002), 63-82. doi: 10.1017/S0022112001007224.
[23]	Z. Lin and Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146. doi: 10.1002/cpa.20239.
[24]	X. Liu, Y. Liu and C. Qu, Stability of peakons for the Novikov equation, J. Math. Pures Appl., 101 (2014), 172-187. doi: 10.1016/j.matpur.2013.05.007.
[25]	Y. Martel, F. Merle and T.-P. Tsai, Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations, Comm. Math. Phys., 231 (2002), 347-373. doi: 10.1007/s00220-002-0723-2.
[26]	A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A, 35 (2002), 4775-4790. doi: 10.1088/0305-4470/35/22/309.
[27]	L. Molinet, A Liouville property with application to asymptotic stability for the Camassa-Holm equation, Arch. Ration. Mech. Anal., 230 (2018), 185-230. doi: 10.1007/s00205-018-1243-3.
[28]	L. Molinet, A rigidity result for the Holm-Staley b-family of equations with application to the asymptotic stability of the Degasperis-Procesi peakon, Nonlinear Anal. Real World Appl., 50 (2019), 675-705. doi: 10.1016/j.nonrwa.2019.06.004.
[29]	L. Molinet, Asymptotic stability for some non positive perturbations of the Camassa-Holm peakon with application to the antipeakon-peakon profile, arXiv: 1804.06230v2
[30]	L. Molinet, On well-posedness results for Camassa-Holm equation on the line: A survey., J. Nonlinear Math. Phys., 11 (2004), 521-533. doi: 10.2991/jnmp.2004.11.4.8.
[31]	V. Novikov, Generalizations of the Camassa-Holm type equation, J. Phys. A, 42 (2009), 342002, 14 pp. doi: 10.1088/1751-8113/42/34/342002.
[32]	J. M. Palacios, Asymptotic stability of peakons for the Novikov equation, J. Differential Equations, 269 (2020), 7750-7791. doi: 10.1016/j.jde.2020.05.039.
[33]	J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48. doi: 10.12775/TMNA.1996.001.
[34]	X. Wu and Z. Yin, Global weak solutions for the Novikov equation., J. Phys. A, 44 (2011), 055202, 17 pp. doi: 10.1088/1751-8113/44/5/055202.