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 & Continuous Dynamical Systems - A, 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. Google Scholar

[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. Google Scholar

[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. 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

[5]

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

[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. Google Scholar

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

[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. 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

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

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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. Google Scholar

[12]

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

[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. Google Scholar

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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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[35]

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

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. Google Scholar

[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. Google Scholar

[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. 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]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[9]

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

[10]

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

[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. Google Scholar

[12]

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

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. Google Scholar

[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. Google Scholar

[18]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[32]

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

[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. Google Scholar

[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. Google Scholar

[35]

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

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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