• Previous Article
    Properties of multicorrelation sequences and large returns under some ergodicity assumptions
  • DCDS Home
  • This Issue
  • Next Article
    Asymptotic behavior of minimal solutions of $ -\Delta u = \lambda f(u) $ as $ \lambda\to-\infty $
doi: 10.3934/dcds.2020372

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

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

Received  May 2020 Revised  September 2020 Published  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.

Citation: José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. M. ChenF. GuoY. 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.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.   Google Scholar

[13]

A. DegasperisD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. N. W. HoneH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. LiuY. 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.  Google Scholar

[25]

Y. MartelF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. M. ChenF. GuoY. 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.  Google Scholar

[6]

A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535.  doi: 10.1007/s00222-006-0002-5.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

R. Danchin, A few remarks on the Camassa-Holm equation, Differential Integral Equations, 14 (2001), 953-988.   Google Scholar

[13]

A. DegasperisD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

A. N. W. HoneH. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. LiuY. 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.  Google Scholar

[25]

Y. MartelF. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

J. F. Toland, Stokes waves, Topol. Methods Nonlinear Anal., 7 (1996), 1-48.  doi: 10.12775/TMNA.1996.001.  Google Scholar

[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.  Google Scholar

[1]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[2]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[3]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[4]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[5]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[6]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[7]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[8]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[9]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[10]

Xavier Carvajal, Liliana Esquivel, Raphael Santos. On local well-posedness and ill-posedness results for a coupled system of mkdv type equations. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020382

[11]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[12]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[13]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[14]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[15]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[16]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[17]

Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318

[18]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[19]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[20]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

2019 Impact Factor: 1.338

Article outline

[Back to Top]