The classical problem of water waves: A reservoir of integrable and nearly-integrable equations

Journal of Nonlinear Mathematical Physics

Volumn 10, Issue , 2003, Pages 72-92

  Johnson, Robin S ^a  

^a NEWCASTLE UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038455796     PISSN: 14029251     EISSN: 17760852     Source Type: Journal    
DOI: 10.2991/jnmp.2003.10.s1.6     Document Type: Article

References (40)

1. Anker, D, and Freeman, N C. 1978. On the Soliton Solutions of the Davey–Stewartson Equations for Long Waves. Proc. Roy. Soc., A360:529–540.
2. Benjamin, T B, and Feir, J E. 1967. The Disintegration of Wave Trains on Deep Water, I:Theory. J. Fluid Mech., 27:417–430.
3. Benney, D J, and Bergeron, R F. 1969. A New Class of Nonlinear Waves in Parallel Flows. Stud. Appl. Math., 48:181–204.
4. Boussinesq, J. 1871. Théorie de l’intumescence liquid appelée onde solitaire ou de translation, se propageant dans un canal rectanulaire. Comptes Rendus Acad. Sci. (Paris), 72:755–759.
5. Burns, J C. 1953. Long Waves in Running Water. Proc. Camb. Phil. Soc., 49:695–706.
6. Camassa, R, and Holm, D D. 1993. An Integrable Shallow Water Equation with Peaked Solitons. Phys. Rev. Lett., 71:1661–1664.
7. Camassa, R, Holm, D D, and Hyman, J M. 1994. A New Integrable Shallow Water Equation. Adv. Appl. Mech., 31:1–33.
8. Constantin, A. 2001. On the Scattering Problem for the Camassa–Holm Equation. Proc. Roy. Soc., A457:953–970.
9. Davey, A, and Stewartson, K. 1974. On Three-Dimensional Packets of Surface Waves. Proc. Roy. Soc., A338:101–110.
10. Degasperis A, Holm D D, Hone A N W, A New Integrable Equation with Peakon Solutions, Theoret. Math. Phys. (2002), to appear
11. Djordjevic, V D, and Redekopp, L G. 1978. On the Development of Packets of Surface Gravity Waves Moving over an Uneven Bottom. J. Appl. Math. Phys., 29:950–962.
12. Drazin P G, Johnson R S, Solitons:An Introduction, Cambridge University Press, 1992
13. Dullin, H R, Gottwald, G, and Holm, D D. 2001. An Integrable Shallow Water Equation with Linear and Nonlinear Dispersion. Phys. Rev. Lett., 87:1945–1948.
14. Fokas, A S, and Fuchssteiner, B. 1981. Symplectic Structures, their Bäcklund Transformations and Hereditary Symmetries. Physica, D4:47–66.
15. Freeman, N C, and Davey, A. 1975. On the Evolution of Packets of Long Surface Waves. Proc. Roy. Soc., A344:427–433.
16. Green G, On the Motion of Waves in a Variable Canal of Small Depth and Width, Camb. Trans.VI (1837) [Papers, p.255]
17. Hasimoto, H, and Ono, H. 1972. Nonlinear Modulation of Gravity Waves. J. Phys. Soc. Japan, 33:805–811.
18. Hietarinta, J. 1987. A Search for Bilinear Equations Passing Hirota’s Three-Soliton Condition. I KdV-Type Bilinear Equations. J. Math. Phys., 28:1732–1742.
19. Johnson, R S. 1976. On the Modulation of Water Waves on Shear Flows. Proc. Roy. Soc., A347:537–546.
20. Johnson, R S. 1977. On the Modulation of Water Waves in the Neighbourhood of kh ≈ 1.363. Proc. Roy. Soc., A357:131–141.
21. Johnson, R S. 1986. On the Nonlinear Critical Layer below a Nonlinear Unsteady Surface Wave. J. Fluid Mech., 167:327–351.
22. Johnson, R S. 1991. 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 Dynamics, 57:115–133.
23. Johnson, R S. 1994. Solitary Wave, Soliton and Shelf Evolution over Variable Depth. J. Fluid Mech., 276:125–138.
24. Johnson, R S. 1996. A Two-Dimensional Boussinesq Equation for Water Waves and Some of Its Solutions. J. Fluid Mech., 323:65–78.
25. Johnson R S, Nonlinear Gravity Waves on the Surface of an Arbitrary Shear Flow with Variable Depth, in Nonlinear Instability Analysis, Editors:Debnath L, Choudhury S R, Comp. Mech. Publications, 1997, 221–243
26. Johnson, R S. 2002. Camassa–Holm, Korteweg-de Vries and Related Models for Water Waves. J. Fluid Mech., 455:63–82.
27. Johnson, R S. 2003. On Solutions of the Camassa–Holm Equation. Proc. R. Soc. Lond., A459:1687–1708.
28. Johnson, R S. 2003. The Camassa–Holm Equation for Water Waves Moving over a Shear Flow. Fluid Dyn. Res., 33(1–2):97–111.
29. Kadomtsev, B P, and Petviashvili, V I. 1970. On the Stability of Solitary Waves in Weakly Dispersing Media. Sov. Phys. Dokl., 15:539–541.
30. Knickerbocker, C J, and Newell, A C. 1980. Shelves and the Korteweg-de Vries Equation. J. Fluid Mech., 98:803–818.
31. Knickerbocker, C J, and Newell, A C. 1985. Reflections from Solitary Waves in Channels of Decreasing Depth. J. Fluid Mech., 153:1–16.
32. Korteweg, D J, and de Vries, G. 1895. On the Change of Form of Long Waves Advancing in a Rectangular Canal, and on a New Type of Long Stationary Waves. Phil. Mag. (5), 39:422–443.
33. Maslowe, S A, and Redekopp, L G. 1980. Long Nonlinear Waves in Stratified Shear Flows. J. Fluid Mech., 101:321–348.
34. Matveev, V B, and Salle, M A. 1991. Darboux Transformations and Solitons, Berlin:Springer-Verlag.
35. Maxon, S, and Viecelli, J. 1974. Cylindrical Solitons. Phys. Fluids, 17:1614–1616.
36. Miles, J W. 1977. Obliquely Interacting Solitary Waves. J. Fluid Mech., 79:157–169.
37. Miles, J W. 1979. On the Korteweg-de Vries Equation for a Gradually Varying Channel. J. Fluid Mech., 91:181–190.
38. Rayleigh, Lord, and Strutt, J W. 1876. On Waves. Phil. Mag. (5), 1:257–279.
39. Russell J S, Report on Waves, in Rep. 14th Meet. Brit. Assoc. Adv. Sci., York, John Murray, London, 1844, 311–390
40. Stokes, G G. 1847. On the Theory of Oscillatory Waves. Trans. Camb. Phil. Soc., 8:161–441.