The Camassa-Holm equation for water waves moving over a shear flow

Fluid Dynamics Research

Volumn 33, Issue 1-2, 2003, Pages 97-111

  Johnson, R S ^a  

^a NEWCASTLE UNIVERSITY   (United Kingdom)

Author keywords

Camassa Holm; Completely integrable; Shear flows; Solitons; Water waves

Indexed keywords

GRAVITY WAVES; SHEAR FLOW; SURFACE WAVES; VORTEX FLOW; WAVE EQUATIONS; WAVE PROPAGATION;

LINEAR SHEAR;

WATER WAVES;

EQUATION; FLOW FIELD; INVISCID FLUID; MATHEMATICAL MODELING; SHEAR FLOW; WATER WAVE;

EID: 0038170200     PISSN: 01695983     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0169-5983(03)00036-4     Document Type: Article

References (24)

1. Beals R. Sattinger D.H. Szmigielski Acoustic scattering and the extended Korteweg-de Vries hierarchy Adv. Math. 140 1998 190-206
2. Benjamin T.B. The solitary wave on a stream with an arbitrary distribution of vorticity J. Fluid Mech. 12 1962 97-116
3. Burns J.C. Long waves on running water Proc. Cambridge Phil. Soc. 49 1953 695-706
4. Busuioc V. On second grade fluids with vanishing viscosity C. R. Acad. Sci. Paris 328 1999 1241-1246
5. Camassa R. Holm D.D. An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1993 1661-1664
6. Camassa R. Holm D.D. Hyman J.M. A new integrable shallow water equation Adv. Appl. Mech. 31 1994 1-33
7. Constantin A. On the scattering problem for the Camassa-Holm equation Proc. R. Soc. London A 457 2001 953-970
8. Constantin A. Strauss W.A. Stability of the Camassa-Holm solitons J. Nonlinear Sci. 12 2002 415-422
9. Dai H.-H. Model equations for nonlinear dispersive waves in a compressible Mooney-Rivlin rod Acta Mech. 127 1998 293-308
10. Drazin P.G. Johnson R.S. Solitons: An Introduction 1992 Cambridge University Press Cambridge
11. Dullin H.R. Gottwald G. Holm D.D. An integrable shallow water equation with linear and nonlinear dispersion Phys. Rev. Lett. 87 2001 1945-1948
12. Fokas A.S. On a class of physically important integrable equations Physica D 87 1995 145-150
13. Fokas A.S. Fuchssteiner B. Symplectic structures, their Bäcklund transformations and hereditary symmetries Physica D 4 1981 47-66
14. Freeman N.C. Johnson R.S. Shallow water waves on shear flows J. Fluid Mech. 42 1970 401-409
15. Johnson R.S. On solutions of the Burns condition (which determines the speed of propagation of linear long waves on a shear flow with or without a critical layer) Geophys. Astrophys. Fluid Dyn. 57 1991 115-133
16. Johnson R.S. A Modern Introduction to the Mathematical Theory of Water Waves 1997 Cambridge University Press Cambridge
17. Johnson R.S. Camassa-Holm, Korteweg-de Vries and related models for water waves J. Fluid Mech. 455 2002 63-82
18. Johnson, R.S., 2002b. On solutions of the Camassa-Holm equation. Proc. R. Soc. London, to appear.
19. Johnson, R.S., 2003. The classical problem of water waves: a reservoir of integrable and nearly-integrable equations J. Nonlinear Math. Phys., to appear (The Öresund Symposium on Partial Differential Equations, 2002).
20. Kodama Y. On integrable systems with higher order corrections Phys. Lett. A 107 1985 245-249
21. Misiolek G. A shallow water equation as a geodesic flow on the Bott-Virasoro group J. Geom. Phys. 24 1998 203-208
22. Thompson P.D. The propagation of small surface disturbances through rotational flow Ann. NY Acad. Sci. 51 1949 463-474
23. Velthuizen H.G.M. van Wijngaarden L. Gravity waves over a non-uniform flow J. Fluid Mech. 39 1969 817-829
24. Yih C.-S. Surface waves in flowing water J. Fluid Mech. 51 1972 209-220