On rotational water waves with surface tension

Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 365, Issue 1858, 2007, Pages 2215-2225

  Wahlén, Erik ^a  

^a LUND UNIVERSITY   (Sweden)

Author keywords

Surface tension; Vorticity; Water waves

Indexed keywords

EID: 34548268647     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2007.2003     Document Type: Article

References (50)

1. Amick, C. J. Kirchgässner, K. 1989 A theory of solitary water waves in the presence of surface tension. Arch. Rat. Mech. Anal. 105, 1-49. (doi:10.1007/BF00251596)
2. Beale, J. T. 1977 The existence of solitary water waves. Comm. Pure Appl. Math. 30, 373-389.
3. Buffoni, B. & Groves, M. D. 1999 A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory. Arch. Rat. Mech. Anal. 146, 183-220. (doi:10.1007/ S002050050141)
4. Buffoni, B., Groves, M. D. & Toland, J. F. 1996 A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers. Phil. Trans. R. Soc. A 354, 575-607. (doi: 10.1098/rsta. 1996.0020)
5. Constantin, A. 2001a Edge waves along a sloping beach. J. Phys. A 34, 9723-9731. (doi:10.1088/ 0305-4470/34/45/311)
6. Constantin, A. 20016 On the deep water wave motion. J. Phys. A 34, 1405-1417. (doi: 10.1088/ 0305-4470/34/7/313)
7. Constantin, A. 2006 The trajectories of particles in Stokes waves. Invent. Math. 166, 523-535. (doi: 10.1007/s00222-006-0002-5)
8. Constantin, A. & Escher, J. 2004a Symmetry of steady deep-water waves with vorticity. Eur. J. Appl. Math. 15, 755-768. (doi:10.1017/S0956792504005777)
9. Constantin, A. & Escher, J. 20046 Symmetry of steady periodic surface water waves with vorticity. J. Fluid Mech. 498, 171-181. (doi: 10.1017/S0022112003006773)
10. Constantin, A. & Strauss, W. 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57, 481-527. (doi:10.1002/cpa.3046)
11. Constantin, A. & Villari, G. In press. Particle trajectories in linear water waves. J. Math. Fluid Mech. (doi:10.1007/s00021-005-0214-2)
12. Constantin, A., Sattinger, D. & Strauss, W. 2006 Variational formulations for steady water waves with vorticity. J. Fluid Mech. 548, 151-163. (doi:10.1017/S0022112005007469)
13. Crandall, M. & Rabinowitz, P. 1971 Bifurcation from simple eigenvalues. J. Fund. Anal. 8, 321-340. (doi:10.1016/0022-1236(71) 90015-2)
14. Crapper, G. D. 1957 An exact solution for progressive capillary waves of arbitrary amplitude. J. Fluid Mech. 2, 532-540. (doi: 10.1017/S0022112057000348)
15. Crapper, G. D. 1984 Introduction to water waves. Chichester, UK: Ellis Horwood Ltd.
16. Dubreil-Jacotin, M. L. 1934 Sur la détermination rigoureuse des ondes permanents périodiques d'ampleur finie. J. Math. Pure Appl. 13, 217-291.
17. Ehrnstöm, M. 2005 Uniqueness for steady periodic water waves with vorticity. Int. Math. Res. Not. 2005, 3721-3726. (doi:10.1155/IMRN.2005.3721)
18. Gerstner, F. 1809 Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412-445.
19. Groves, M. D. & Wahlen, E. 2006 Spatial dynamics methods for solitary gravity-capillary water waves with an arbitrary distribution of vorticity. Preprint.
20. Hogan, S. J. 1984 Particle trajectories in nonlinear capillary waves. J. Fluid Mech. 143, 243-252. (doi:10.1017/S0022112084001336)
21. Hogan, S. J, 1985 Particle trajectories in nonlinear gravity-capillary waves. J. Fluid Mech. 151, 105-119. (doi:10.1017/ S002211208500088X)
22. Hogan, S. J. 1986 Highest waves, phase speeds and particle trajectories of nonlinear capillary waves on sheets of fluid. J. Fluid Mech. 172, 547-563. (doi:10.1017/S0022112086001866)
23. Hur, V. M. 2006 Global bifurcation theory of deep-water waves with vorticity. SIAM J. Math. Anal. 37, 1482-1521. (doi:10.1137/ 040621168)
24. Hur, V. M. In press. Exact solitary water waves with vorticity. Arch. Rat. Mech. Anal.
25. Iooss, G. & Kirchgässner, K. 1990 Bifurcation d'ondes solitaires en présence d'une faible tension superficielle. CR. Acad. Sci. Paris 311, 265-268.
26. Iooss, G. & Kirchgässner, K. 1992 Water waves for small surface tension: an approach via normal form. Proc. R. Soc. Edin. A 122, 267-299.
27. Iooss, G. & Pérouème, M. C. 1993 Perturbed homoclinic solutions in reversible 1 : 1 resonance vector fields. J. Differ. Equations 102, 62-88. (doi:10.1006/jdeq.1993.1022)
28. Johnson, R. S. 1997 A modern, introduction to the mathematical theory of water waves. Cambridge, UK: Cambridge University Press.
29. Jones, M. 1989 Small amplitude capillary-gravity waves in a channel of finite depth. Glasgow Math. J. 31, 141-160.
30. Jones, M. & Toland, J. 1985 The bifurcation and secondary bifurcation of capillary-gravity waves. Proc. R. Soc. A 399, 391-417. (doi:10.1098/rspa.1985.0063)
31. Jones, M. & Toland, J. 1986 Symmetry and the bifurcation of capillary-gravity waves. Arch. Rat. Mech. Anal. 96, 29-53. (doi:10.1007/BF00251412)
32. Kalisch, H. 2004 A uniqueness result for periodic traveling waves in water of finite depth. Nonlinear Anal. 58, 779-785. (doi:10.1016/j.na.2004.01.008)
33. Kinnersley, W. 1976 Exact large amplitude capillary waves on sheets of fluid. J. Fluid Mech. 77, 229-241. (doi:10.1017/S0022112076002085)
34. Kirchgässner, K. 1982 Wave solutions of reversible systems and applications. J. Differ. Equations 45, 113-127. (doi: 10.1016/0022-0396(82)90058-4)
35. Kirchgässner, K. 1988 Nonlinearly resonant surface waves and homoclinic bifurcation. Adv. Appl. Mech. 26, 135-181.
36. Lighthill, J. 1978 Waves in fluids. Cambridge, UK: Cambridge University Press.
37. Luo, Y. & Trudinger, N. 1991 Linear second order elliptic equations with Venttsel boundary conditions. Proc. R. Soc. Edin. A 118, 193-207.
38. Mielke, A. 1991 Hamiltonian and Lagrangian flows on center manifolds. Berlin, Germany: Springer.
39. Okamoto, H. & Shoji, M. 2001 The mathematical theory of permanent progressive water-waves. River Edge, NJ: World Scientific.
40. Reeder, J. & Shinbrot, M. 1981a On Wilton ripples. I. Formal derivation of the phenomenon. Wave Motion 3, 115-135. (doi: 10.1016/0165-2125(81)90001-9)
41. Reeder, J. & Shinbrot, M. 19816 On Wilton ripples. II. Rigorous results. Arch. Rat. Mech. Anal. 77, 321-347. (doi:10.1007/ BF00280641)
42. Sachs, R. L. 1991 On the existence of small amplitude solitary waves with strong surface tension. J. Differ. Equations 90, 31-51. (doi:10.1016/0022-0396(91)90159-7)
43. Da Silva, T. & Peregrine, D. H. 1988 Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281-302. (doi:10.1017/S0022112088002423)
44. Swan, C., Cummins, I. & James, R. 2001 An experimental study of two-dimensional surface water waves propagating on depth-varying currents. J. Fluid Mech. 428, 273-304. (doi:10.1017/ S0022112000002457)
45. Toland, J. F. 1996 Stokes waves. Topol. Methods Nonlinear Anal. 7, 1-48.
46. Wahlén, E. 2005 A note on steady gravity waves with vorticity. Int. Math. Res. Not. 7, 389-396. (doi:10.1155/IMRN.2005.389)
47. Wahlen, E. 2006a Steady periodic capillary-gravity waves with vorticity. SIAM J. Math. Anal. 38, 921-943. (doi: 10.1137/050630465)
48. Wahlen, E. 2006b Steady periodic capillary waves with vorticity. Ark. Mat. 44, 367-387. (doi:10. 1007/S11512-006-0024-7)
49. Wilton, J. R. 1915 On ripples. Phil. Mag. 29, 688-700.
50. Zeidler, E. 1973 Existenzbeweis für permanente Kapillar- Schwerewellen mit allgemeinen Wirbelverteilungen. Arch. Rat. Mech. Anal. 50, 34-72. (doi:10.1007/BF00251293)