We propose a hierarchy of nonlinearly dispersive generalized Korteweg–de Vries (KdV) evolution equations based on a modification of the Lagrangian density whose induced action functional the KdV equation extremizes. It is shown that two recent nonlinear evolution equations describing wave propagation in certain generalized continua with an inherent material length scale are members of the proposed hierarchy. Like KdV, the equations from the proposed hierarchy possess Hamiltonian structure. Unlike KdV, however, the solutions to these equations can be compact (i.e., they vanish outside of some open interval) and, in addition, peaked. Implicit solutions for these peaked, compact traveling waves (“peakompactons”) are presented. Korteweg–de Vries equation, compact solitary waves, classical field theory, Lagrangian mechanics, Hamiltonian mechanics.
1. Boussinesq, J. Essai sur la théorie des eaux courantes. Mémoires préséntes par divers savants à l’Académie des Sciences de l’Institut de France, XXIII:1–680, 1877.
2. Korteweg, D. J. and de Vries, G. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Phil. Mag. (Ser. 5)}, 1895, 39, 422–443.
3. Zabusky, N. J. and Kruskal, M. D. Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 1965, 15, 240–243.
http://dx.doi.org/10.1103/PhysRevLett.15.240
4. Miura, R. M. The Korteweg–de Vries equation: A survey of results. SIAM Rev., 1976, 18, 412–459.
http://dx.doi.org/10.1137/1018076
5. Jeffrey, A. and Kakutani, T. Weak nonlinear dispersive waves: A discussion centered around the Korteweg–de Vries equation. SIAM Rev., 1972, 14, 582–643.
http://dx.doi.org/10.1137/1014101
6. Miura, R. M. Korteweg–de Vries equation and generalizations. I. A remarkable explicit nonlinear transformation. J. Math. Phys., 1968, 9, 1202–1204.
http://dx.doi.org/10.1063/1.1664700
7. Johnson, R. S. On the inverse scattering transform, the cylindrical Korteweg–De Vries equation and similarity solutions. Phys. Lett. A, 1979, 72, 197–199.
http://dx.doi.org/10.1016/0375-9601(79)90002-1
8. Marchant, T. R. and Smyth, N. F. The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography. J. Fluid Mech., 1990, 221, 263–287.
http://dx.doi.org/10.1017/S0022112090003561
9. Camassa, R. and Holm, D. D. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 1993, 71, 1661–1664.
http://dx.doi.org/10.1103/PhysRevLett.71.1661
10. Rosenau, P. and Hyman, J. M. Compactons: Solitons with finite wavelength. Phys. Rev. Lett., 1993, 70, 564–567.
http://dx.doi.org/10.1103/PhysRevLett.70.564
11. Cooper, F., Shepard, H., and Sodano, P. Solitary waves in a class of generalized Korteweg–de Vries equations. Phys. Rev. E, 1993, 48, 4027–4032.
http://dx.doi.org/10.1103/PhysRevE.48.4027
12. Christov, C. I. and Velarde, M. G. Dissipative solitons. Physica D, 1995, 86, 323–347.
http://dx.doi.org/10.1016/0167-2789(95)00111-G
13. Fokas, A. S. On a class of physically important integrable equations. Physica D, 1995, 87, 145–150.
http://dx.doi.org/10.1016/0167-2789(95)00133-O
14. Salupere, A., Engelbrecht, J., and Maugin, G. A. Solitonic structures in KdV-based higher-order systems. Wave Motion, 2001, 34, 51–61.
http://dx.doi.org/10.1016/S0165-2125(01)00069-5
15. Maugin, G. A. Solitons in elastic solids (1938–2010). Mech. Res. Commun., 2011, 38, 341–349.
http://dx.doi.org/10.1016/j.mechrescom.2011.04.009
16. Engelbrecht, J. and Pastrone, F. Waves in microstructured solids with strong nonlinearities in microscale. Proc. Estonian Acad. Sci. Phys. Math., 2003, 52, 12–20.
17. Randrüüt, M., Salupere, A., and Engelbrecht, J. On modelling wave motion in microstructured solids. Proc. Estonian Acad. Sci., 2009, 58, 241–246.
http://dx.doi.org/10.3176/proc.2009.4.05
18. Salupere, A., Lints, M., and Engelbrecht, J. On solitons in media modelled by the hierarchical KdV equation. Arch. Appl. Mech., 2014, 9–11, 1583–1593.
http://dx.doi.org/10.1007/s00419-014-0861-y
19. Destrade, M. and Saccomandi, G. Solitary and compact-like shear waves in the bulk of solids. Phys. Rev. E, 2006, 73, 065604.
http://dx.doi.org/10.1103/PhysRevE.73.065604
20. Jordan, P. M. and Saccomandi, G. Compact acoustic travelling waves in a class of fluids with nonlinear material dispersion. Proc. R. Soc. A, 2012, 468, 3441–3457.
http://dx.doi.org/10.1098/rspa.2012.0321
21. Rosenau, P. and Oron, A. On compactons induced by a non-convex convection. Commun. Nonlinear Sci. Numer. Simulat., 2014, 19, 1329–1337.
http://dx.doi.org/10.1016/j.cnsns.2013.09.028
22. Rubin, M. B., Rosenau, P., and Gottlieb, O. Continuum model of dispersion caused by an inherent material characteristic length. J. Appl. Phys., 1995, 77, 4054–4063.
http://dx.doi.org/10.1063/1.359488
23. Gardner, C. S. Korteweg–de Vries equation and generalizations. IV. The Korteweg–de Vries equation as a Hamiltonian system. J. Math. Phys., 1971, 12, 1548–1551.
http://dx.doi.org/10.1063/1.1665772
24. Gelfand, I. M. and Fomin, S. V. Calculus of Variations. Dover Publications, Mineola, NY, 2000.
25. Maugin, G. A. and Christov, C. I. Nonlinear duality between elastic waves and quasi-particles. In Selected Topics in Nonlinear Wave Mechanics (Christov, C. I. and Guran, A., eds). Birkhäuser, Boston, 2002, 117–160.
http://dx.doi.org/10.1007/978-1-4612-0095-6_4
26. Kruskal, M. D. and Zabusky, N. J. Exact invariants for a class of nonlinear wave equations. J. Math. Phys., 1966, 7, 1256–1267.
http://dx.doi.org/10.1063/1.1705028
27. Holm, D. D., Schmah, T., and Stoica, C. Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford University Press, New York, 2009.
28. Olde Daalhuis, A. B. Hypergeometric function. In NIST Digital Library of Mathematical Functions (Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W., eds). Chapter 15, http://dlmf.nist.gov/, release 1.0.9, 2014.
29. Li, Y. A. and Olver, P. J. Convergence of solitary-wave solutions in a perturbed bi-Hamiltonian dynamical system I. Compactions and peakons. Discret. Contin. Dyn. S. A, 1997, 3, 419–432.
http://dx.doi.org/10.3934/dcds.1997.3.419
