American Institute of Mathematical Sciences

Advanced
May  2021, 41(5): 2475-2518. doi: 10.3934/dcds.2020372

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  May 2020 Revised  September 2020 Published  May 2021 Early access  November 2020

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.

Keywords: Solitons, stability, asymptotic stability, peakons, Camassa-Holm type equation, Liouville-type theorems.
Mathematics Subject Classification: 35Q35, 35Q51, 35C08, 35B35, 37K40.
Citation: José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2475-2518. doi: 10.3934/dcds.2020372
References:
[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.

show all references

References:
[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.

[1]

Aiyong Chen, Xinhui Lu. Orbital stability of elliptic periodic peakons for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1703-1735. doi: 10.3934/dcds.2020090
[2]

Byungsoo Moon. Orbital stability of periodic peakons for the generalized modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4409-4437. doi: 10.3934/dcdss.2021123
[3]

Xingxing Liu. Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5505-5521. doi: 10.3934/dcds.2018242
[4]

Yongsheng Mi, Boling Guo, Chunlai Mu. On an $N$-Component Camassa-Holm equation with peakons. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1575-1601. doi: 10.3934/dcds.2017065
[5]

Quoc Hung Phan. Optimal Liouville-type theorems for a parabolic system. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 399-409. doi: 10.3934/dcds.2015.35.399
[6]

Yongsheng Mi, Chunlai Mu. On a three-Component Camassa-Holm equation with peakons. Kinetic and Related Models, 2014, 7 (2) : 305-339. doi: 10.3934/krm.2014.7.305
[7]

Stephen Anco, Daniel Kraus. Hamiltonian structure of peakons as weak solutions for the modified Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4449-4465. doi: 10.3934/dcds.2018194
[8]

Kaouther Ammar, Philippe Souplet. Liouville-type theorems and universal bounds for nonnegative solutions of the porous medium equation with source. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 665-689. doi: 10.3934/dcds.2010.26.665
[9]

Pavol Quittner, Philippe Souplet. Parabolic Liouville-type theorems via their elliptic counterparts. Conference Publications, 2011, 2011 (Special) : 1206-1213. doi: 10.3934/proc.2011.2011.1206
[10]

Marianna Euler, Norbert Euler. Integrating factors and conservation laws for some Camassa-Holm type equations. Communications on Pure and Applied Analysis, 2012, 11 (4) : 1421-1430. doi: 10.3934/cpaa.2012.11.1421
[11]

Zeng Zhang, Zhaoyang Yin. On the Cauchy problem for a four-component Camassa-Holm type system. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5153-5169. doi: 10.3934/dcds.2015.35.5153
[12]

Phuong Le. Liouville theorems for an integral equation of Choquard type. Communications on Pure and Applied Analysis, 2020, 19 (2) : 771-783. doi: 10.3934/cpaa.2020036
[13]

Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063
[14]

Alberto Farina. Symmetry of components, Liouville-type theorems and classification results for some nonlinear elliptic systems. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 5869-5877. doi: 10.3934/dcds.2015.35.5869
[15]

Philippe Souplet. Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices. Networks and Heterogeneous Media, 2012, 7 (4) : 967-988. doi: 10.3934/nhm.2012.7.967
[16]

Yongsheng Mi, Boling Guo, Chunlai Mu. Persistence properties for the generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1623-1630. doi: 10.3934/dcdsb.2019243
[17]

Yu Gao, Jian-Guo Liu. The modified Camassa-Holm equation in Lagrangian coordinates. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2545-2592. doi: 10.3934/dcdsb.2018067
[18]

Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
[19]

Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
[20]

Defu Chen, Yongsheng Li, Wei Yan. On the Cauchy problem for a generalized Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 871-889. doi: 10.3934/dcds.2015.35.871

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (256)
  • HTML views (184)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy