On asymptotically equivalent shallow water wave equations

Physica D: Nonlinear Phenomena

Volumn 190, Issue 1-2, 2004, Pages 1-14

  Dullin, H R ^a,b   Gottwald, G A ^c   Holm, D D ^d,e  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

^b UNIVERSITY OF BREMEN   (Germany)

^c UNIVERSITY OF SYDNEY   (Australia)

^d LOS ALAMOS NATIONAL LABORATORY   (United States)

^e IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Asymptotics; Korteweg de Vries equation; Water wave equations

Indexed keywords

APPROXIMATION THEORY; DISPERSION (WAVES); INTEGRATION; MATHEMATICAL MODELS; MATHEMATICAL TRANSFORMATIONS; NONLINEAR EQUATIONS; PROBLEM SOLVING; SURFACE TENSION; VELOCITY MEASUREMENT; WATER WAVES;

ASYMPTOTICS; KORTEWEG-DE VRIES EQUATION; WATER WAVE EQUATIONS;

WAVE EQUATIONS;

EID: 1142304402     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2003.11.004     Document Type: Article

References (38)

1. Alber M., Camassa R., Holm D.D., Marsden J.E., The geometry of peaked solitons and billiard solutions of a class of integrable PDE's. Lett. Math. Phys. 32:1994;137-151.
2. Alber M., Camassa R., Fedorov V.N., Holm D.D., Marsden J.E., On billiard solutions of nonlinear PDE's. Phys. Lett. A. 264:1999;171-178 Alber M., Camassa R., Fedorov V.N., Holm D.D., Marsden J.E., The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type. Comm. Math. Phys. 221:2001;197-227.
3. Alber M., Camassa R., Fedorov V.N., Holm D.D., Marsden J.E., On billiard solutions of nonlinear PDE's. Phys. Lett. A. 264:1999;171-178 Alber M., Camassa R., Fedorov V.N., Holm D.D., Marsden J.E., The complex geometry of weak piecewise smooth solutions of integrable nonlinear PDE's of shallow water and Dym type. Comm. Math. Phys. 221:2001;197-227.
4. Benjamin T.B., Bona J.L., Mahoney J.J., Model equations for long waves in nonlinear dispersive systems. Phil. Trans. Roy. Soc. Lond. A. 272:1972;47-78.
5. Beale T.J., Solitary water waves with capillary ripples at infinity. Commun. Pure Appl. Math. 64:1991;211-257.
6. Camassa R., Holm D.D., An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71:1993;1661-1664.
7. Champneys A.R., Groves M.D., A global investigations of solitary-wave solutions of a two-parameter model for water waves. J. Fluid Mech. 342:1997;199-229.
8. Champneys A.R., Vanden-Broeck J.-M., Lord G.J., Do true elevation solitary waves exist? A numerical investigation. J. Fluid Mech. 454:2002;403-417.
9. Craig W., Groves M.D., Hamiltonian long-wave approximations to the water-wave problem. Wave Motion. 19:1994;367-389.
10. A. Degasperis, M. Procesi, Asymptotic integrability, in: A. Degasperis, G. Gaeta (Eds.), Symmetry and Perturbation Theory, World Scientific, Singapore, 1999, pp. 23-37.
11. A. Degasperis, D.D. Holm, A.N.W. Hone, A new integrable equation with peakon solutions, Theoret. Math. Phys. 133 (2002) 1463-1474. http://xxx.lanl.gov/abs/nlin.SI/0205023.
12. Dias F., Kharif C., Nonlinear gravity and capillary-gravity waves. Ann. Rev. Fluid Mech. 31:1999;301-346.
13. Dullin H.R., Gottwald G., Holm D.D., An integrable shallow water equation with linear and nonlinear dispersion. Phys. Rev. Lett. 87:2001;4501-4504.
14. Fenton J.D., A ninth-order solution for the solitary wave. J. Fluid Mech. 53:1972;257-271.
15. J.D. Fenton, Nonlinear wave theories, in: B. Le Méhauté, D.M. Hanes (Eds.), The Sea: Ocean Engineering Science, Part A, vol. 9, 1990, pp. 3-25.
16. Fokas A.S., Liu Q.M., Asymptotic integrability of water waves. Phys. Rev. Lett. 77:1996;2347-2351.
17. Fokas A.S., On a class of physically important integrable equations. Physica D. 87:1995;145-150.
18. Fokas A.S., Fuchssteiner B., Bäcklund transformations for hereditary symmetries. Nonlinear Anal. TMA. 5:1981;423-432.
19. Fuchssteiner B., Some tricks from the symmetry-toolbox for nonlinear equations: generalization of the Camassa-Holm equation. Physica D. 95:1996;229-243.
20. Grimshaw R.H.J., The solitary wave in water of variable depth, Part 2. J. Fluid Mech. 46:1971;611-622.
21. Grimshaw R.H.J., Joshi N., Weakly nonlocal solitary waves in a singularly perturbed Korteweg-de Vries equation. SIAM J. Appl. Math. 55:1995;124-135.
22. Holm D.D., Marsden J.E., Ratiu T.S., The Euler-Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137:1998;1-81.
23. D.D. Holm, M.F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Phys. Lett. A 308 (2003) 437-444 http://xxx.lanl.gov/abs/nlin.CD/0203007 D.D. Holm, M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dynam. Syst. 2 (2003) 323-380.
24. D.D. Holm, M.F. Staley, Nonlinear balance and exchange of stability in dynamics of solitons, peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE, Phys. Lett. A 308 (2003) 437-444 http://xxx.lanl.gov/abs/nlin.CD/0203007 D.D. Holm, M.F. Staley, Wave structure and nonlinear balances in a family of evolutionary PDEs, SIAM J. Appl. Dynam. Syst. 2 (2003) 323-380.
25. Johnson R.S., Camassa-Holm, Korteweg-de Vries and related models for water waves. J. Fluid Mech. 455:2002;63-82.
26. Kodama Y., On integrable systems with higher order corrections. Phys. Lett. A. 107:1985;245-249 Y. Kodama, Normal forms for weakly dispersive wave equations, 112 (1985) 193-196 Kodama Y., On solitary-wave interaction. Phys. Lett. A. 123:1987;276-282.
27. Kodama Y., On integrable systems with higher order corrections. Phys. Lett. A. 107:1985;245-249 Y. Kodama, Normal forms for weakly dispersive wave equations, 112 (1985) 193-196 Kodama Y., On solitary-wave interaction. Phys. Lett. A. 123:1987;276-282.
28. Kodama Y., On integrable systems with higher order corrections. Phys. Lett. A. 107:1985;245-249 Y. Kodama, Normal forms for weakly dispersive wave equations, 112 (1985) 193-196 Kodama Y., On solitary-wave interaction. Phys. Lett. A. 123:1987;276-282.
29. Y. Kodama, A.V. Mikhailov, Obstacles to asymptotic integrability, in: A.S. Fokas, I.M. Gelfand (Eds.), Algebraic Aspects of Integrable Systems: In Memory of Irene Dorfman, Birkhäuser, Boston, 1996, pp. 173-204.
30. Korteweg D.J., de Vries G., On the change of form of long waves advancing in a rectangular channel, and a new type of long stationary waves. Phil. Mag. 39(5):1895;422-443.
31. Li Z., Sibgatullin N.R., An improved theory of long waves on the water surface. J. Appl. Math. Mech. 61:1997;177-182.
32. E. Lombardi, Oscillatory integrals and phenomena beyond all algebraic orders. With applications to homoclinic orbits in reversible system, Lecture Notes in Mathematics, vol. 1741, Springer-Verlag, Berlin, 2000.
33. Marchant T.R., Smyth N.F., The extended Korteweg-de Vries equation and the resonant flow over topography. J. Fluid Mech. 221:1990;263-288.
34. Mikhailov A.V., Novikov V.S., Perturbative symmetry approach. J. Phys. A. 35:2002;4775-4790.
35. Olver P.J., Hamiltonian perturbation theory and water waves. Contemp. Math. 28:1984;231-249.
36. Sun S.M., Non-existence of truly solitary waves in water with small surface tension. Proc. Roy. Soc. Lond. A. 455:1999;2191-2228.
37. G.B. Whitham, Linear and Nonlinear Waves, Wiley/Interscience, New York, 1974.
38. Weidman P.D., Maxworthy T., Experiments on strong interactions between solitary waves. J. Fluid Mech. 85:1978;417-431.