A bifurcation theory for three-dimensional oblique travelling gravity-capillary water waves

Journal of Nonlinear Science

Volumn 13, Issue 4, 2003, Pages 397-447

  Groves, M D ^a   Haragus, M ^b  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

^b UNIVERSITÉ DE FRANCHE COMTÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84867952271     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-003-0530-8     Document Type: Article

References (33)

1. AKYLAS, T. 1994. Three-dimensional long water-wave phenomena. Ann. Rev. Fluid Mech. 26, 191-210. (Pubitemid 24987422)
2. AMBROSETTI, A., AND PRODI, G. 1995. A Primer of Nonlinear Analysis, paperback ed. Cambridge: C.U.P. (Cambridge Studies in Advanced Mathematics 34).
3. BUFFONI, B., AND GROVES, M. D. 1999. A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory. Arch. Ratl. Mech. Anal. 146, 183-220.
4. BUFFONI, B.,GROVES.M.D., AND TOLAND, J. F. 1996.Aplethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers. Phil. Trans. Roy. Soc. London A 354, 575-607.
5. CRAIG.W., AND NICHOLLS, D. P. 2000. Traveling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal. 32, 323-359. (Pubitemid 33239152)
6. DEBOUARD, A., AND SAUT, J.-C. 1997. Solitarywaves of generalized Kadomtsev-Petviashvili equations. Ann. Inst. H. Poincaŕe Anal. Non Lińeaire 14, 211-236.
7. DIAS, F., AND IOOSS, G. 2003. Water-waves as a spatial dynamical system. In Handbook of Mathematical Fluid Dynamics, vol. 2 (ed. Friedlander, S. and Serre, D.), pages 443-499. Amsterdam: North-Holland.
8. DIAS, F., AND KHARIF, C. 1999. Nonlinear gravity and capillary-gravity waves. Ann. Rev. Fluid Mech. 31, 301-346. (Pubitemid 36178536)
9. DUISTERMAAT, J. J. 1984. Bifurcation of periodic solutions near equilibrium points of Hamiltonian systems. In Lecture Notes in Mathematics 1057-Bifurcation Theory and Applications, Montecatini 1983 (ed. Salvadori, L.), pages 57-105. New York: Springer-Verlag.
10. GROVES, M. D. 2001. An existence theory for three-dimensional periodic travelling gravitycapillary water waves with bounded transverse profiles. Physica D 152-153, 395-415.
11. GROVES, M. D., AND MIELKE, A. 2001. A spatial dynamics approach to three-dimensional gravity-capillary steady water waves. Proc. Roy. Soc. Edinburgh A 131, 83-136. (Pubitemid 33773577)
12. HAMMACK, J., SCHEFFNER, N., AND SEGUR, H. 1989. Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567-589. (Pubitemid 20635461)
13. HARAGUS-COURCELLE, M., AND KIRCHG̈ASSNER, K. 2001. Three-dimensional steady capillary-gravity waves. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. Fiedler, B.), pages 363-397. Berlin: Springer-Verlag.
14. HARAGUS-COURCELLE, M., AND PEGO, R. L. 2000. Spatial wave dynamics of steady oblique wave interactions. Physica D 145, 207-232.
15. IOOSS, G., AND KIRCHG̈ASSNER, K. 1990. Bifurcation d'ondes solitaires en pŕesence d'une faible tension superficielle. C. R. Acad. Sci. Paris, Śer. 1 311, 265-268.
16. IOOSS, G., ANDKIRCHG̈ASSNER, K. 1992.Waterwaves for small surface tension: An approach via normal form. Proc. Roy. Soc. Edinburgh A 122, 267-299.
17. IOOSS, G., AND P'EROÙEME, M. C. 1993. Perturbed homoclinic solutions in reversible 1:1 resonance vector fields. J. Diff. Eq. 102, 62-88.
18. JONES, M. C.W. 1989. Small amplitude gravity-capillary waves in a channel of finite depth. Glasgow Math. J. 31, 141-160.
19. KIRCHG̈ASSNER, K. 1982.Wave solutions of reversible systems and applications. J. Diff. Eq. 45, 113-127.
20. KIRCHG̈ASSNER, K. 1988. Nonlinear resonant surfacewaves and homoclinic bifurcation. Adv. Appl. Math. 26, 135-181.
21. LOMBARDI, E. 1997. Orbits homoclinic to exponentially small periodic orbits for a class of reversible systems. Application to water waves. Arch. Ratl. Mech. Anal. 137, 227-304.
22. LOMBARDI, E. 2000. Oscillatory Integrals and Phenomena beyond All Algebraic Orders. Berlin: Springer-Verlag.
23. LUKE, J. C. 1967. A variational principle for a fluid with a free surface. J. Fluid Mech. 27, 395-397.
24. MIELKE, A. 1988. Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sci. 10, 51-66.
25. MIELKE, A. 1991. Hamiltonian and Lagrangian Flows on Center Manifolds. Berlin: Springer-Verlag.
26. MOSER, J. 1976. Periodic orbits near an equilibrium and a theorem by Alan Weinstein. Commun. Pure Appl. Math. 29, 727-747.
27. PEGO, R. L., AND QUINTERO, J. 2002. A host of traveling waves in a model of threedimensional water-wave dynamics. J. Nonlinear Sci. 12, 59-83. (Pubitemid 135698356)
28. PLOTNIKOV, P. I. 1980. Solvability of the problem of spatial gravitationalwaves on the surface of an ideal fluid. Sov. Phys. Dokl. 25, 170-171.
29. REEDER, J., AND SHINBROT, M. 1981. Three-dimensional nonlinear wave interaction in water of constant depth. Nonlinear Analysis TMA 5, 303-323.
30. SCHMIDT, D. S. 1974. Periodic solutions near a resonant equilibrium of a Hamiltonian system. Celest. Mech. 9, 81-103.
31. STOKER, J. J. 1957. Water Waves: The Mathematical Theory with Applications. New York: Interscience.
32. WEINSTEIN, A. 1973. Normal modes for nonlinear Hamiltonian systems. Invent. Math. 20, 47-57.
33. WILTON, J. R. 1915. On ripples. Phil. Mag. 29, 688-700.