Secondary criticality of water waves. Part 1. Definition, bifurcation and solitary waves

Journal of Fluid Mechanics

Volumn 565, Issue , 2006, Pages 381-417

  Bridges, T J ^a   Donaldson, N M ^a  

^a UNIVERSITY OF SURREY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

FLOW OF WATER; GRAVITY WAVES; MATHEMATICAL MODELS; STEADY FLOW; UNSTEADY FLOW;

NON-UNIFORM FLOWS; SHALLOW WATER; SOLITARY WAVES; STOKES WAVE; UNIFORM FLOWS;

WATER WAVES;

FLOW OF WATER; GRAVITY WAVES; MATHEMATICAL MODELS; STEADY FLOW; UNSTEADY FLOW; WATER WAVES;

HYDRAULICS; SOLITARY WAVE; UNSTEADY FLOW; WATER WAVE;

EID: 33749353307     PISSN: 00221120     EISSN: 14697645     Source Type: Journal    
DOI: 10.1017/S002211200600187X     Document Type: Article

References (61)

1. Abbott, M. B. 1979 Computational Hydraulics. Pitman.
2. Amick, C. J. & Toland, J. F. 1981 On periodic water waves and their convergence to solitary waves in the long-wave limit. Phil. Trans. R. Soc. Lond. A 303, 633-669.
3. Amick, C. & Toland, J. F. 1992 Solitary waves with surface tension I: trajectories homoclinic to periodic orbits in four dimensions. Arch. Rat. Mech. Anal. 118, 127-148.
4. Baesens, C. & MacKay, R. S. 1992 Uniformly travelling water waves from a dynamical systems viewpoint: Some insights into bifurcations from Stokes' family. J. Fluid Mech. 241, 333-347.
5. Benjamin, T. B. 1971 A unified theory of conjugate flows. Phil. Trans. R. Soc. Lond. A 269, 587-643.
6. Benjamin, T. B. 1984 Impulse, flow force and variational principles. IMA J. Appl. Maths 32, 3-68.
7. Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448-460.
8. Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137-185.
9. Boussinesq, J. 1877 Essai sur la théorie des eaux courantes. Memoires presentes par divers savant a l'Academie des Science de l'Institut de France 23, 1-680.
10. Bridges T. J. 1992 Spatial Hamiltonian structure, energy flux and the water-wave problem. Proc. R. Soc. Lond. A 439, 297-315.
11. Bridges T. J. 1996 Periodic patterns, linear instability, symplectic structure and mean-flow dynamics for three-dimensional surface waves. Phil. Trans. R. Soc. Lond. A 354, 533-574.
12. Bridges T. J. 1997a Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc. 121, 147-190.
13. Bridges T. J. 1997b A geometrical formulation of the conservation of wave action and its implications for signature and the classification of instabilities. Proc. R. Soc. Lond. A 453, 1365-1395.
14. Bridges T. J. 2001 Transverse instability of solitary-wave states of the water-wave problem. J. Fluid Mech. 439, 255-278.
15. Bridges T. J. 2004 Superharmonic instability, homoclinic torus bifurcation and water-wave breaking. J. Fluid Mech. 505, 153-162.
16. Bridges T. J. 2005 Steady dark solitary waves emerging from wave-generated meanflow: The role of modulation equations. Chaos 15, 037113.
17. Bridges T. J. 2006a Canonical multi-symplectic structure on the total exterior algebra bundle. Proc. R. Soc. Lond. A 462, 1531-1551.
18. Bridges T. J. 2006b Degenerate relative equilibria, curvature of the momentum map, and homoclinic bifurcation. Preprint, University of Surrey.
19. Bridges T. J. & Donaldson, N. M. 2005 Degenerate periodic orbits and homoclinic torus bifurcation Phys. Rev. Lett. 95, 104301.
20. Bridges T. J. & Donaldson, N. M. 2006 Secondary criticality of water waves. Part 2. Unsteady criticality and the Benjamin-Feir instability from the viewpoint of hydraulics. J. Fluid Mech. 565, 419-439.
21. Bridges T. J. & Fan, E. G. 2004 Solitary waves, periodic waves, and a stability analysis for Zufiria's higher-order Boussinesq model for shallow water waves. Phys. Lett. A 326, 381-390.
22. Bridges T. J. & Mielke, A. 1995 A proof of the Benjamin-Feir instability. Arch. Rat. Mech. Anal. 133, 145-198
23. Bridges T. J., Christodoulides, P. & Dias, F. 1995 Spatial bifurcations of capillary-gravity interfacial waves when the phase and group velocities are nearly equal. J. Fluid Mech. 295, 121-158.
24. Buffoni, B., Dancer, E. N. & Toland, J. F. 2000 The subharmonic bifurcation of Stokes waves. Arch. Rat. Mech. Anal. 152, 241-271.
25. Champneys, A. R. & Lord, G. J. 1997 Computation of homoclinic solutions to periodic orbits in a reduced water-wave problem. Physica D 102, 101-124.
26. Christov, C. I. 2001 An energy-consistent dispersive shallow-water model. Wave Motion 34, 161-174.
27. Cokelet, E. D. 1977 Steep gravity waves in water of arbitrary uniform depth. Phil. Trans. R. Soc. Lond. A 286, 183-230.
28. Craig, W. & Groves, M. D. 1994 Hamiltonian long-wave approximations to the water-wave problem. Wave Motion 19, 367-389.
29. Deymie, P. H. 1935 Propagation d'une intumescence allongée. Rev. Gén. L'Hydraulique 1, 138-142.
30. Dias, F. & Iooss, G. 1994 Ondes solitaires 'noires' à l'interface entre deux fluides en presence de tension superficielle. C. R. Acad. Sci. Paris Sér. I Math. 319, 89-93.
31. Dias, F. & Iooss, G. 2003 Water waves as a spatial dynamical system. Handbook of Mathematical Fluid Dynamics, vol. 2, pp. 443-499. North-Holland.
32. Dingemans, M. W. 1997 Water Wave Propagation over Uneven Bottoms. Part 2 - Non-linear Wave Propagation. World Scientific.
33. Dooge, J. C. I. 1987 Historical development of concepts in open channel flow. In Hydraulics and Hydraulic Research: A Historical Review (ed. G. Garbrecht, A. A. Balkema. pp. 205-230.
34. Doole, S. H. 1997 The bifurcation of long waves in the parameter space of pressure head and flowforce. Q. J. Mech. Appl. Maths 50, 17-34.
35. Doole, S. H. & Norbury, J. 1995 The bifurcation of steady gravity water waves in (R, S) parameter space. J. Fluid Mech. 302, 287-305.
36. Gill, A. E. 1977 The hydraulics of rotating channel flow. J. Fluid Mech. 80, 641-671.
37. 4. J. Fluid Mech. 405, 181-210.
38. Grimshaw, R. H. J. & Joshi, N. 1995 Weakly nonlinear solitary waves in a singularly-perturbed Korteweg-deVries equation. SIAM J. Appl. Maths 55, 124-135.
39. Groves, M. D. & Toland, J. F. 1997 On variational formulations for steady water waves. Arch. Rat. Mech. Anal. 137, 203-226.
40. Henderson, F. M. 1966 Open Channel Flow. Macmillan.
41. Iooss, G. & Adelmeyer, M. 1992 Topics in Bjfurcation Theory and Applications. World Scientific.
42. Johnson, E. R. & Clarke, S. R. 2001 Rossby wave hydraulics. Annu. Rev. Fluid Mech. 33, 207-230.
43. Keady, G. & Norbury, J. 1975 Water waves and conjugate streams. J. Fluid Mech. 70, 663-671.
44. Killworth, P. D. 1992 On hydraulic control in a stratified fluid. J. Fluid Mech. 237, 605-626.
45. Lancaster, P. & Tismenetsky, M. 1985 The Theory of Matrices 2nd edn., Academic.
46. Luke, J. C. 1967 A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395-397.
47. Madsen, P. A., Bingham, H. B. & Liu, H. 2002 A new Boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 1-30.
48. Magnus, J. R. & Neudecker, H. 1988 Matrix Differential Calculus. John Wiley.
49. Marsden, J. E. 1992 Lectures on Mechanics. London Mathematical Society. vol. 174. Cambridge University Press.
50. McLean, J. W. 1982 Instabilities of finite-amplitude gravity waves on water of finite depth. J. Fluid Mech. 114, 331-341.
51. Mielke, A. 1991 Hamiltonian and Lagrangian Flows on Center-Manifolds with Applications to Elliptic Variational Problems. Lecture Notes in Maths. vol. 1489. Springer.
52. Mielke, A. 1996 A spatial center-manifold approach to steady bifurcations from spatially periodic patterns. Pitman Res. Notes Maths 352, 209-277.
53. Nutku, Y. 1983 Canonical formulations of shallow water flows. J. Phys. A: Maths Gen. 16, 4195-4201.
54. Peregrine, D. H. 1983 Water waves, nonlinear Schrödinger equations and their solutions. J. Austral. Math. Soc. B 25, 16-43.
55. Peregrine, D. H. 1985 Water waves and their development in space and time. Proc. R. Soc. Lond. A 400, 1-18.
56. Roberts, A. J. & Peregrine, D. H. 1983 Notes on long-crested water waves. J. Fluid Mech. 135, 323-335.
57. Stiassne, M. & Peregrine, D. H. 1980 Shoaling of finite-amplitude surface waves on water of slowly-varying depth. J. Fluid Mech. 97, 783-805.
58. Vandenbroeck, J.-M. 1983 Some new gravity waves in water of finite depth. Phys. Fluids 26, 2385-2387.
59. Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.
60. Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. Zh. Prikl. Mekh. Fiz. 9, 86-94 (English trans: J. Appl. Mech. Tech. Phys. 2, 190).
61. Zufiria, J. A. 1987 Weakly nonlinear non-symmetric gravity waves on water of finite depth. J. Fluid Mech. 180, 371-385.