Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An $ ab$-family of equations with peakon traveling waves


Authors: A. Alexandrou Himonas and Dionyssios Mantzavinos
Journal: Proc. Amer. Math. Soc. 144 (2016), 3797-3811
MSC (2010): Primary 35Q53, 37K10, 37C07
DOI: https://doi.org/10.1090/proc/13011
Published electronically: February 12, 2016
MathSciNet review: 3513539
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Peakon traveling wave solutions, both on the line and on the circle, are derived for a novel $ ab$-family of nonlocal evolution equations with cubic nonlinearities. At least two members of this $ ab$-family, namely the Fokas-Olver-Rosenau-Qiao equation and the Novikov equation, are known to be integrable. Furthermore, a generalization of the $ ab$-family with nonlinearities of order $ k\in \mathbb{N}$, $ k\geqslant 2$, is considered and its multi-peakon on the line is obtained.


References [Enhancements On Off] (What's this?)

  • [B] J. V. Boussinesq, Essai sur la théorie des eaux courantes, Mémoires présentés par divers savants à l'Académie des Sciences 23 (1877), no. 1, 1-680.
  • [By] Peter Byers, Existence time for the Camassa-Holm equation and the critical Sobolev index, Indiana Univ. Math. J. 55 (2006), no. 3, 941-954. MR 2244592 (2007k:35395), https://doi.org/10.1512/iumj.2006.55.2710
  • [CH] Roberto Camassa and Darryl D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett. 71 (1993), no. 11, 1661-1664. MR 1234453 (94f:35121), https://doi.org/10.1103/PhysRevLett.71.1661
  • [CL] Adrian Constantin and David Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal. 192 (2009), no. 1, 165-186. MR 2481064 (2010f:35334), https://doi.org/10.1007/s00205-008-0128-2
  • [F] A. S. Fokas, On a class of physically important integrable equations, Phys. D 87 (1995), no. 1-4, 145-150. The nonlinear Schrödinger equation (Chernogolovka, 1994). MR 1361680, https://doi.org/10.1016/0167-2789(95)00133-O
  • [Fu] Benno Fuchssteiner, Some tricks from the symmetry-toolbox for nonlinear equations: generalizations of the Camassa-Holm equation, Phys. D 95 (1996), no. 3-4, 229-243. MR 1406283 (97f:35184 ), https://doi.org/10.1016/0167-2789(96)00048-6
  • [FF1] A. S. Fokas and B. Fuchssteiner, On the structure of symplectic operators and hereditary symmetries, Lett. Nuovo Cimento (2) 28 (1980), no. 8, 299-303. MR 580058 (82g:58039)
  • [FF2] B. Fuchssteiner and A. S. Fokas, Symplectic structures, their Bäcklund transformations and hereditary symmetries, Phys. D 4 (1981/82), no. 1, 47-66. MR 636470 (84j:58046), https://doi.org/10.1016/0167-2789(81)90004-X
  • [GGKM] C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), no. 19, 1095-1097.
  • [GLOQ] Guilong Gui, Yue Liu, Peter J. Olver, and Changzheng Qu, Wave-breaking and peakons for a modified Camassa-Holm equation, Comm. Math. Phys. 319 (2013), no. 3, 731-759. MR 3040374, https://doi.org/10.1007/s00220-012-1566-0
  • [HG] Katelyn Grayshan and A. Alexandrou Himonas, Equations with peakon traveling wave solutions, Adv. Dyn. Syst. Appl. 8 (2013), no. 2, 217-232. MR 3162143
  • [HH1] A. Alexandrou Himonas and Curtis Holliman, On well-posedness of the Degasperis-Procesi equation, Discrete Contin. Dyn. Syst. 31 (2011), no. 2, 469-488. MR 2805816 (2012g:35289), https://doi.org/10.3934/dcds.2011.31.469
  • [HH2] A. Alexandrou Himonas and Curtis Holliman, The Cauchy problem for the Novikov equation, Nonlinearity 25 (2012), no. 2, 449-479. MR 2876876 (2012k:35469), https://doi.org/10.1088/0951-7715/25/2/449
  • [HH3] A. Alexandrou Himonas and Curtis Holliman, The Cauchy problem for a generalized Camassa-Holm equation, Adv. Differential Equations 19 (2014), no. 1-2, 161-200. MR 3161659
  • [HHG] A. Alexandrou Himonas, Curtis Holliman, and Katelyn Grayshan, Norm inflation and ill-posedness for the Degasperis-Procesi equation, Comm. Partial Differential Equations 39 (2014), no. 12, 2198-2215. MR 3259553, https://doi.org/10.1080/03605302.2014.942737
  • [HMa] A. Alexandrou Himonas and Dionyssios Mantzavinos, The Cauchy problem for the Fokas-Olver-Rosenau-Qiao equation, Nonlinear Anal. 95 (2014), 499-529. MR 3130540, https://doi.org/10.1016/j.na.2013.09.028
  • [HMa2] A. Alexandrou Himonas and Dionyssios Mantzavinos, The Cauchy problem for a 4-parameter family of equations with peakon traveling waves, Nonlinear Anal. 133 (2016), 161-199. MR 3449754, https://doi.org/10.1016/j.na.2015.12.012
  • [HMi] A. Alexandrou Himonas and Gerard Misiołek, The Cauchy problem for an integrable shallow-water equation, Differential Integral Equations 14 (2001), no. 7, 821-831. MR 1828326 (2002c:35228)
  • [HS1] Darryl D. Holm and Martin F. Staley, Nonlinear balance and exchange of stability of dynamics of solitons, peakons, ramps/cliffs and leftons in a $ 1+1$ nonlinear evolutionary PDE, Phys. Lett. A 308 (2003), no. 5-6, 437-444. MR 1977364 (2004c:35345), https://doi.org/10.1016/S0375-9601(03)00114-2
  • [HS2] Darryl D. Holm and Martin F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dyn. Syst. 2 (2003), no. 3, 323-380 (electronic). MR 2031278 (2004k:76046), https://doi.org/10.1137/S1111111102410943
  • [HLS] Andrew N. W. Hone, Hans Lundmark, and Jacek Szmigielski, Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation, Dyn. Partial Differ. Equ. 6 (2009), no. 3, 253-289. MR 2569508 (2010i:37172), https://doi.org/10.4310/DPDE.2009.v6.n3.a3
  • [HW] Andrew N. W. Hone and Jing Ping Wang, Integrable peakon equations with cubic nonlinearity, J. Phys. A 41 (2008), no. 37, 372002, 10. MR 2430566 (2009i:35311), https://doi.org/10.1088/1751-8113/41/37/372002
  • [L] Jonatan Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations 217 (2005), no. 2, 393-430. MR 2168830 (2007k:35417), https://doi.org/10.1016/j.jde.2004.09.007
  • [LO] Yi A. Li and Peter J. Olver, Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation, J. Differential Equations 162 (2000), no. 1, 27-63. MR 1741872 (2002a:35185), https://doi.org/10.1006/jdeq.1999.3683
  • [MN] A. V. Mikhailov and V. S. Novikov, Perturbative symmetry approach, J. Phys. A 35 (2002), no. 22, 4775-4790. MR 1908645 (2004d:35012), https://doi.org/10.1088/0305-4470/35/22/309
  • [N1] Vladimir Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A 42 (2009), no. 34, 342002, 14. MR 2530232 (2011b:35466), https://doi.org/10.1088/1751-8113/42/34/342002
  • [N2] V. Novikov, Personal communication. (2015).
  • [OR] Peter J. Olver and Philip Rosenau, Tri-Hamiltonian duality between solitons and solitary-wave solutions having compact support, Phys. Rev. E (3) 53 (1996), no. 2, 1900-1906. MR 1401317 (97c:35172), https://doi.org/10.1103/PhysRevE.53.1900
  • [Q] Zhijun Qiao, A new integrable equation with cuspons and W/M-shape-peaks solitons, J. Math. Phys. 47 (2006), no. 11, 112701, 9. MR 2278659 (2007i:37127), https://doi.org/10.1063/1.2365758
  • [T] F. Tığlay, The periodic Cauchy problem for Novikov's equation, Int. Math. Res. Not. IMRN 20 (2011), 4633-4648. MR 2844933, https://doi.org/10.1093/imrn/rnq267
  • [W] G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. Pure and Applied Mathematics. MR 0483954 (58 #3905)
  • [ZK] N. J. Zabusky and M. D. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett. 15 (1965), no. 6, 240-243.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 35Q53, 37K10, 37C07

Retrieve articles in all journals with MSC (2010): 35Q53, 37K10, 37C07


Additional Information

A. Alexandrou Himonas
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: himonas.1@nd.edu

Dionyssios Mantzavinos
Affiliation: Department of Mathematics, State University of New York at Buffalo, Buffalo, New York 14260
Email: dionyssi@buffalo.edu

DOI: https://doi.org/10.1090/proc/13011
Keywords: Fokas, Olver, Rosenau, Qiao equation, Novikov equation, Camassa-Holm equation, Degasperis-Procesi equation, $b$-family of equations, integrable equations, peakon, multi-peakon, conserved quantities.
Received by editor(s): August 19, 2015
Received by editor(s) in revised form: October 24, 2015
Published electronically: February 12, 2016
Communicated by: Catherine Sulem
Article copyright: © Copyright 2016 American Mathematical Society

American Mathematical Society