October  2019, 39(10): 6131-6148. doi: 10.3934/dcds.2019267

Accelerating dynamical peakons and their behaviour

1. 

Department of Mathematics and Statistics, Brock University, St. Catharines, ON L2S3A1, Canada

2. 

Department of Mathematics, Faculty of Sciences, Universidad de Cádiz, Puerto Real, Cádiz, 11510, Spain

Received  March 2019 Revised  May 2019 Published  July 2019

A wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illustrate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others.

Citation: Stephen C. Anco, Elena Recio. Accelerating dynamical peakons and their behaviour. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 6131-6148. doi: 10.3934/dcds.2019267
References:
[1]

S. C. Anco and F. Mobasheramini, Integrable U(1)-invariant peakon equations from the NLS hierarchy, Physica D, 355 (2017), 1-23.  doi: 10.1016/j.physd.2017.06.006.

[2]

S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A.: Math. Theor., 52 (2019), 125203. doi: 10.1088/1751-8121/ab03dd.

[3]

S. C. Anco, P. L. da Silva and I. L. Freire, A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations, J. Math. Phys., 56 (2015), 091506 (21pp). doi: 10.1063/1.4929661.

[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]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.

[6]

A. Constantin and W. 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.

[7]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. 

[8]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[9]

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

[10]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.

[11]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[12]

A. Degasperis and M. Procesi, Asymptotic integrability, In: Proc. Symmetry and Perturbation Theory(Rome, 1998), 23–37. World Sci. Publ., 1999.

[13]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422.

[14]

J. EscherY. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485.  doi: 10.1016/j.jfa.2006.03.022.

[15]

A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.  doi: 10.1007/BF00994638.

[16]

A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations., In: Algebraic Aspects of Integrable Systems, 93–101. Progr. Nonlinear Differential Equations Appl., vol. 26, Brikhauser Boston, 1997.

[17]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.

[18]

G. Grayshan and A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. and Appl., 8 (2013), 217–232.

[19]

G. GuiY. LiuP. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731-759.  doi: 10.1007/s00220-012-1566-0.

[20]

A. Himonas and D. Mantzavinos, An ab-family of equations with peakon travelling waves, Proc. Amer. Math. Soc., 144 (2016), 3797-3811.  doi: 10.1090/proc/13011.

[21]

A. Himonas and D. Mantzavinos, The Cauchy problem for a 4-parameter family of equations with peakon travelling waves, Nonlin. Anal., 133 (2016), 161-199.  doi: 10.1016/j.na.2015.12.012.

[22]

D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonl. Math. Phys., 12 (2005), 380-394.  doi: 10.2991/jnmp.2005.12.s1.31.

[23]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312.  doi: 10.2991/jnmp.2007.14.3.1.

[24]

Z. Jiang and L. Ni, Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl., 385 (2012), 551-558.  doi: 10.1016/j.jmaa.2011.06.067.

[25]

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.

[26]

J. Lenells, A variational approach to the stability of periodic peakons, J. Nonl. Math. Phys., 11 (2004), 151-163.  doi: 10.2991/jnmp.2004.11.2.2.

[27]

Y. Liu and Z. Lin, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.  doi: 10.1002/cpa.20239.

[28]

Y. LiuP. J. OlverC. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Analysis Appl., 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.

[29]

Y. Mi and C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Diff. Equ., 254 (2013), 961-982.  doi: 10.1016/j.jde.2012.09.016.

[30]

V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys A: Math. Theor., 42 (2009), 342002 (14 pp). doi: 10.1088/1751-8113/42/34/342002.

[31]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev., 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.

[32]

P. J. Olver, Classical Invariant Theory, Cambridge University Press (Cambridge, UK), 1999. doi: 10.1017/CBO9780511623660.

[33]

C. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonliearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.

[34]

E. Recio and S. C. Anco, Conserved norms and related conservation laws for multi-peakon equations, J. Phys. A: Math. Theor., 51 (2018), 065203 (19pp). doi: 10.1088/1751-8121/aaa0e0.

[35]

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

show all references

References:
[1]

S. C. Anco and F. Mobasheramini, Integrable U(1)-invariant peakon equations from the NLS hierarchy, Physica D, 355 (2017), 1-23.  doi: 10.1016/j.physd.2017.06.006.

[2]

S. C. Anco and E. Recio, A general family of multi-peakon equations and their properties, J. Phys. A.: Math. Theor., 52 (2019), 125203. doi: 10.1088/1751-8121/ab03dd.

[3]

S. C. Anco, P. L. da Silva and I. L. Freire, A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations, J. Math. Phys., 56 (2015), 091506 (21pp). doi: 10.1063/1.4929661.

[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]

A. Constantin and J. Escher, Wave breaking for nonlinear nonlocal shallow water equations, Acta Math., 181 (1998), 229-243.  doi: 10.1007/BF02392586.

[6]

A. Constantin and W. 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.

[7]

A. Constantin and J. Escher, Global existence and blow-up for a shallow water equation, Ann. Scuola Norm. Sup. Pisa, 26 (1998), 303-328. 

[8]

A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal., 192 (2009), 165-186.  doi: 10.1007/s00205-008-0128-2.

[9]

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

[10]

A. Constantin, Particle trajectories in extreme Stokes waves, IMA J. Appl. Math., 77 (2012), 293-307.  doi: 10.1093/imamat/hxs033.

[11]

A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc., 44 (2007), 423-431.  doi: 10.1090/S0273-0979-07-01159-7.

[12]

A. Degasperis and M. Procesi, Asymptotic integrability, In: Proc. Symmetry and Perturbation Theory(Rome, 1998), 23–37. World Sci. Publ., 1999.

[13]

A. DegasperisD. D. Holm and A. N. W. Hone, A new integrable equation with peakon solutions, Theor. Math. Phys., 133 (2002), 1463-1474.  doi: 10.1023/A:1021186408422.

[14]

J. EscherY. Liu and Z. Yin, Global weak solutions and blow-up structure for the Degasperis-Procesi equation, J. Funct. Anal., 241 (2006), 457-485.  doi: 10.1016/j.jfa.2006.03.022.

[15]

A. Fokas, The Korteweg-de Vries equation and beyond, Acta Appl. Math., 39 (1995), 295-305.  doi: 10.1007/BF00994638.

[16]

A. S. Fokas, P. J. Olver and P. Rosenau, A plethora of integrable bi-Hamiltonian equations., In: Algebraic Aspects of Integrable Systems, 93–101. Progr. Nonlinear Differential Equations Appl., vol. 26, Brikhauser Boston, 1997.

[17]

B. Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D, 95 (1996), 229-243.  doi: 10.1016/0167-2789(96)00048-6.

[18]

G. Grayshan and A. Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. and Appl., 8 (2013), 217–232.

[19]

G. GuiY. LiuP. J. Olver and C. Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Commun. Math. Phys., 319 (2013), 731-759.  doi: 10.1007/s00220-012-1566-0.

[20]

A. Himonas and D. Mantzavinos, An ab-family of equations with peakon travelling waves, Proc. Amer. Math. Soc., 144 (2016), 3797-3811.  doi: 10.1090/proc/13011.

[21]

A. Himonas and D. Mantzavinos, The Cauchy problem for a 4-parameter family of equations with peakon travelling waves, Nonlin. Anal., 133 (2016), 161-199.  doi: 10.1016/j.na.2015.12.012.

[22]

D. D. Holm and A. N. W. Hone, A class of equations with peakon and pulson solutions, J. Nonl. Math. Phys., 12 (2005), 380-394.  doi: 10.2991/jnmp.2005.12.s1.31.

[23]

D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys., 14 (2007), 303-312.  doi: 10.2991/jnmp.2007.14.3.1.

[24]

Z. Jiang and L. Ni, Blow-up phenomenon for the integrable Novikov equation, J. Math. Anal. Appl., 385 (2012), 551-558.  doi: 10.1016/j.jmaa.2011.06.067.

[25]

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.

[26]

J. Lenells, A variational approach to the stability of periodic peakons, J. Nonl. Math. Phys., 11 (2004), 151-163.  doi: 10.2991/jnmp.2004.11.2.2.

[27]

Y. Liu and Z. Lin, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62 (2009), 125-146.  doi: 10.1002/cpa.20239.

[28]

Y. LiuP. J. OlverC. Qu and S. Zhang, On the blow-up of solutions to the integrable modified Camassa-Holm equation, Analysis Appl., 12 (2014), 355-368.  doi: 10.1142/S0219530514500274.

[29]

Y. Mi and C. Mu, On the Cauchy problem for the modified Novikov equation with peakon solutions, J. Diff. Equ., 254 (2013), 961-982.  doi: 10.1016/j.jde.2012.09.016.

[30]

V. S. Novikov, Generalizations of the Camassa-Holm equation, J. Phys A: Math. Theor., 42 (2009), 342002 (14 pp). doi: 10.1088/1751-8113/42/34/342002.

[31]

P. J. Olver and P. Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev., 53 (1996), 1900-1906.  doi: 10.1103/PhysRevE.53.1900.

[32]

P. J. Olver, Classical Invariant Theory, Cambridge University Press (Cambridge, UK), 1999. doi: 10.1017/CBO9780511623660.

[33]

C. QuX. C. Liu and Y. Liu, Stability of peakons for an integrable modified Camassa-Holm equation with cubic nonliearity, Comm. Math. Phys., 322 (2013), 967-997.  doi: 10.1007/s00220-013-1749-3.

[34]

E. Recio and S. C. Anco, Conserved norms and related conservation laws for multi-peakon equations, J. Phys. A: Math. Theor., 51 (2018), 065203 (19pp). doi: 10.1088/1751-8121/aaa0e0.

[35]

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

Figure 1.  Asymptotic travelling-wave peakon
Figure 2.  Direction-reversing peakon
Figure 3.  Dissipating peakon
Figure 4.  Blowing up peakon
Figure 5.  Peakon breather
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