On the energetic stability of solitary water waves

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

Volumn 360, Issue 1799, 2002, Pages 2337-2358

  Mielke, Alexander ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Dirichlet Neumann operator; Energetic stability; Gravity capillary water waves; Hamiltonian form; Zakharov's formulation

Indexed keywords

WATER;

ARTICLE; ENERGY TRANSFER; FLOW KINETICS; GRAVITY; METHODOLOGY; MOTION; PERIODICITY; SURFACE TENSION; THEORETICAL MODEL; WATER FLOW;

ENERGY TRANSFER; GRAVITATION; MODELS, THEORETICAL; MOTION; PERIODICITY; RHEOLOGY; SURFACE TENSION; WATER; WATER MOVEMENTS;

EID: 0037107586     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2002.1067     Document Type: Article

References (42)

1. Amick, C. J. & Kirchgässner, K. 1989 A theory of solitary water-waves in the presence of surface tension. Arch. Ration. Mech. Analysis 105, 1-49.
2. Arnold, V. 1966 Sur un principe variationel pour les écoulements stationnaires des liquides parfaits et ses applications aux problèmes de stabilité non linéaires. J. Mec. 5, 29-43.
3. 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.
4. Benjamin, T. B. 1972 The stability of solitary waves. Proc. R. Soc. Lond. A 328, 153-183.
5. Benjamin, T. B. & Olver, P. J. 1982 Hamiltonian structure, symmetries and conservation laws for water waves. J. Fluid Mech. 125, 137-185.
6. Beyer, K. & Günther, M. 1998 On the Cauchy problem for a capillary drop. I. Irrotational motion. Math. Meth. Appl. Sci. 21, 1149-1183.
7. Beyer, K., Gawriljuk, I., Günther, M., Lukovsky, I. & Timokha, A. 2001 Compressible potential flows with free boundaries. I. Vibrocapillary equilibria. Z. Angew. Math. Mech. 81, 261-271.
8. Bridges, T. J. 1992 Spatial Hamiltonian structure, energy flux and the water-wave problem. Proc. R. Soc. Lond. A 439, 297-315.
9. Bridges, T. J. 1997 Multi-symplectic structures and wave propagation. Math. Proc. Camb. Phil. Soc. 121, 147-190.
10. Bridges, T. J. & Mielke, A. 1995 A proof of the Benjamin-Feir instability. Arch. Ration. Mech. Analysis 133, 145-198.
11. Buffoni, B., Dancer, E. N. & Toland, J. F. 2000a The regularity and local bifurcation of steady periodic water waves. Arch. Ration. Mech. Analysis 152, 207-240.
12. Buffoni, B., Dancer, E. N. & Toland, J. F. 2000b The sub-harmonic bifurcation of Stokes waves. Arch. Ration. Mech. Analysis 152, 241-271.
13. Calvo, D. C., Yang, T.-S. & Akylas, T. R. 2000 On the stability of solitary waves with decaying oscillatory tails. Proc. R. Soc. Lond. A 456, 469-487.
14. Coifman, R. & Meyer, Y. 1985 Nonlinear harmonic analysis and analytic dependence. In Pseudodifferential operators and applications, Proc. Symp. in Pure Mathematics, Notre Dame, IN, 1984, vol. 43, pp. 71-78. Providence, RI: American Mathematical Society.
15. Costabel, M. 1988 Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Analysis 19, 613-626.
16. Craig, W. 1985 An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. PDEs 10, 787-1003.
17. Craig, W. & Nicholls, D. P. 2000 Traveling two and three dimensional capillary gravity water waves. SIAM J. Math. Analysis 32, 323-359.
18. Craig, W. & Sulem, C. 1993 Numerical simulation of gravity waves. J. Computat. Phys. 108, 73-83.
19. Grillakis, M., Shatah, J. & Strauss, W. 1987 Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Analysis 74, 160-197.
20. Grisvard, P. 1985 Elliptic problems in nonsmooth domains. Boston, MA: Pitman.
21. 1,ρ-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Annln 283, 679-687.
22. Groves, M. D. & Mielke, A. 2001 A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc. R. Soc. Edinb. A 131, 83-136.
23. Groves, M. D. & Toland, J. F. 1997 On variational formulations for steady water waves. Arch. Ration. Mech. Analysis 137, 203-226.
24. Haragus, M. & Scheel, A. 2002 Finite-wavelength stability of capillary-gravity solitary waves. Commun. Math. Phys. 225, 487-521.
25. Iooss, G. & Kirchgässner, K. 1992 Water waves for small surface tension: An approach via normal form. Proc. R. Soc. Edinb. A 122, 267-299.
26. Iooss, G., Lombardi, E. & Sun, S. 2002 Gravity travelling waves for two superposed fluid layers, one being of infinite depth: A new type of bifurcation. Phil. Trans. R. Soc. Lond. A 360, 2245-2336.
27. Kirchgässner, K. 1982 Wave-solutions of reversible systems and applications. J. Diff. Eqns 45, 113-127.
28. Kirchgässner, K. 1988 Nonlinearly resonant surface waves and homoclinic bifurcation. Adv. Appl. Mech. 26, 135-181.
29. Kruse, H.-P. & Scheurle, J. 1998 On the bifurcation and stability of rigidly rotating inviscid liquid bridges. J. Nonlin. Sci. 8, 215-232.
30. Lewis, D. 1989 Nonlinear stability of a rotating planar liquid drop. Arch. Ration. Mech. Analysis 106, 287-333.
31. Lewis, D., Marsden, J. & Ratiu, T. 1987 Stability and bifurcation of a rotating planar liquid drop. J. Math. Phys. 28, 2508-2515.
32. Mielke, A. 1988 Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sci. 10, 51-66.
33. Mielke, A. 1991 Hamiltonian and Lagrangian Flows on Center Manifolds with Applications to Elliptic Variational Problems. Lecture Notes in Mathematics, vol. 1489. Springer.
34. Nicholls, D. P. & Reitich, F. 2001 Stability of high-order perturbative methods for the computation of Dirichlet-Neumann operators. J. Computat. Phys. 170, 276-298.
35. Plotnikov, P. I. & Toland, J. F. 2001 Nash-Moser theory for standing water waves. Arch. Ration. Mech. Analysis 159, 1-83.
36. Schneider, G. & Wayne, C. E. 2000 The long wave limit for the water wave problem. I. The case of zero surface tension. Commun. Pure Appl. Math. 53, 1475-1535.
37. Simo, J. C., Lewis, D. & Marsden, J. E. 1991 Stability of relative equilibria. I. The reduced energy-momentum method. Arch. Ration. Mech. Analysis 115, 15-59.
38. Toland, J. F. 1996 Stokes waves. Topolog. Meth. Nonlin. Analysis 7, 1-48.
39. Wu, S. J. 1993 Analytic dependence of Riemann mappings for bounded domains and minimal surfaces. Commun. Pure Appl. Math. 46, 1303-1326.
40. Wu, S. 1997 Well-posedness in Sobolev spaces of the full water wave problem in 2-D. Invent. Math. 130, 39-72.
41. Wu, S. 1999 Well-posedness in Sobolev spaces of the full water wave problem in 3-D. J. Am. Math. Soc. 12, 445-495.
42. Zakharov, V. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190-194.