May  2021, 41(5): 2475-2518. 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  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.

Citation: José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete & 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.  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

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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

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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

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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

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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

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

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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

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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

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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

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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

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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

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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

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