PL EN


Preferences help
enabled [disable] Abstract
Number of results
2015 | 70 |
Article title

Problems with energy of waves described by Korteweg – de Vries equation

Content
Title variants
Languages of publication
EN
Abstracts
EN
Different forms of the Korteweg -- de Vries equation and their invariants are presented. Different formulas for the energy  of the system described by KdV equation are compared to each other for fixed and moving coordinate systems. It is shown that the energy conservation holds only in moving coordinate systems.
Keywords
EN
 
Year
Volume
70
Physical description
Dates
published
2015
online
29 - 04 - 2016
Contributors
References
  • 1. J. Boussinesq, Essai sur la theorie des eaux courantes, Memoires presentes par divers savants ‘A l’Acad. des Sci. Inst. Nat. France, XXIII, 1877.
  • 2. D.J. Korteveg and G. de Vries, On the change of form of the long waves advancing in a rectangular canal, and on a new type of stationary waves, Phil. Mag. (5), 39, 422 (1895).
  • 3. A. Karczewska, P. Rozmej and Ł. Rutkowski, A new nonlinear equation in the shallow water wave problem, Physica Scripta 89 (2014) 054026.
  • 4. A. Karczewska, P. Rozmej and E. Infeld, Shallow-water soliton dynamics beyond the Korteweg–de Vries equation, Physical Review E 90, 012907 (2014).
  • 5. E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, 2nd edition, Cambridge University Press, 2000.[6] T.R. Marchant and N.F. Smyth, The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography, J. Fluid Mech. (1990), 221, 263-288.
  • 7. G.I. Burde, A. Sergyeyev, Ordering of two small parameters in the shallow water wave problem, J. Phys. A: Math. Theor. 46, 075501 (2013).
  • 8. M. Remoissenet, Waves Called Solitons: Concepts and Experiments, Springer, Berlin, 1999.
  • 9. A. Ali and H. Kalisch, On the formulation of mass, momentum and energy conservation in the KdV equation, Acta Appl. Math. (2014) 133; 113-131.
  • 10. P. Lax, Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Applied Math. 21 (5), (1968) 467–490.
  • 11. C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, Method for Solving the Korteweg-deVries Equation, Phys. Rev. Lett. 19, (1967) 1095-1097.
  • 12. M. Ablowitz, H. Segur, Solitons and the Inverse Scattering Transform, SIAM, Philadelphia, 1981.
  • 13. M. Ablowitz, P. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.
  • 14. P.G. Drazin and R.S. Johnson, Solitons: An Introduction, Cambridge University Press, Cambridge, 1989.
  • 15. R.M. Miura, KdV equation and generalizations I. A remarkable explicit nonlinear transformation, J. Math. Phys. 9 (1968) 1202-1204.
  • 16. R.M. Miura, C.S. Gardner and M.D. Kruskal, KdV equation and generalizations II. Existence of onservation laws and constants of motion, J. Math. Phys. 9 (1968) 1204-1209.
  • 17. J.C. Luke, A variational principle for a fluid with a free surface, J. Fluid Mech. (1967), 27, part 2, 395-397.
  • 18. G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974.
Document Type
Publication order reference
Identifiers
YADDA identifier
bwmeta1.element.ojs-doi-10_17951_aaa_2015_70_43
JavaScript is turned off in your web browser. Turn it on to take full advantage of this site, then refresh the page.