A dimension-breaking phenomenon in the theory of steady gravity-capillary water waves

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

Volumn 360, Issue 1799, 2002, Pages 2189-2243

  Groves, M D ^a   Haragus, M ^a   Sun, S M ^a  

^a LOUGHBOROUGH UNIVERSITY   (United Kingdom)

Author keywords

Hamiltonian systems; Infinite dimensional Lyapunov centre theorem; Three dimensional solitary waves

Indexed keywords

WATER;

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

GRAVITATION; MODELS, THEORETICAL; MOTION; RHEOLOGY; SURFACE TENSION; WATER; WATER MOVEMENTS;

EID: 0037107482     PISSN: 1364503X     EISSN: None     Source Type: Journal    
DOI: 10.1098/rsta.2002.1066     Document Type: Article

References (27)

1. Ablowitz, M. J. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92, 691-715.
2. 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.
3. Bridges, T. J. & Derks, G. 2000 Dimension Breaking of Nonlinear Elliptic PDEs: Satisfying the Spectral Condition Geometrically. Preprint.
4. Buffoni, B. & Groves, M. D. 1999 A multiplicity result for solitary gravity-capillary waves in deep water via critical-point theory. Arch. Ration. Mech. Analysis 146, 183-220.
5. 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. Lond. A 354, 575-607.
6. Devaney, R. L. 1976 Reversible diffeomorphisms and flows. Trans. Am. Math. Soc. 218, 89-113.
7. Dias, F. & Haragus-Courcelle, M. 2000 On the transition from two-dimensional to three-dimensional water waves. Stud. Appl. Math. 104, 91-127.
8. Grisvard, P. 1967 Charactérisation de quelques espaces d'interpolation. Arch. Ration. Mech. Analysis 25, 40-63.
9. Groves, M. D. 2001 An existence theory for three-dimensional periodic travelling gravity-capillary water waves with bounded transverse profiles. Physica D 152-153, 395-415.
10. 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.
11. Haragus, M. & Kirchgässner, K. 1995 Breaking the dimension of a steady wave: Some examples. In Nonlinear Dynamics and Pattern Formation in the Natural Environment (ed. A. Doelman & A. van Harten). Pitman Research Notes in Mathematics Series, vol. 335, pp. 119-129.
12. Haragus-Courcelle, M. & Kirchgässner, K. 2001 Three-dimensional steady capillary-gravity waves. In Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems (ed. B. Fiedler), pp. 363-397. Springer.
13. Haragus-Courcelle, M. & Pego, R. L. 1999 Travelling waves of KP equations with transverse modulations. C. R. Acad. Sci. Paris Sér. I 328, 227-232.
14. Iooss, G. 1999 Gravity and capillary-gravity periodic travelling waves for two superposed fluid layers, one being of infinite depth. J. Math. Fluid Mech. 1, 24-63.
15. Iooss, G. & Kirchgässner, K. 1990 Bifurcation d'ondes solitaires en présence d'une faible tension superficielle. C. R. Acad. Sci. Paris Sér. I 311, 265-268.
16. Iooss, G. & Kirchgässner, K. 1992 Water waves for small surface tension: an approach via normal form. Proc. R. Soc. Edinb. A 122, 267-299.
17. Kadomtsev, B. B. & Petviashvili, V. I. 1970 On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15, 539-541.
18. Kato, T. 1976 Perturbation Theory for Linear Operators, 2nd Edn. Springer.
19. Kirchgässner, K. 1982 Wave solutions of reversible systems and applications. J. Diff. Eqns. 45, 113-127.
20. Kirchgässner, K. 1988 Nonlinear resonant surface waves and homoclinic bifurcation. Adv. Appl. Math. 26, 135-181.
21. Kirchgässner, K. & Schaaf, R. 1994 Homoclinic Bifurcation from Fold Points for Nonlinear Elliptic Problems on Cylindrical Domains. Preprint.
22. Korteweg, D. J. & de Vries, G. 1895 On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves. Phil. Mag. 39, 422-443.
23. Mielke, A. 1988 Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Meth. Appl. Sci. 10, 51-66.
24. Sachs, R. L. 1991 On the existence of small amplitude solitary waves with strong surface-tension. J. Diff. Eqns 90, 31-51.
25. Stoker, J. J. 1957 Water Waves: The Mathematical Theory With Applications. New York: Interscience.
26. Tajiri, M. & Murakami, Y. 1990 The periodic solution resonance: solutions of the Kadomtsev-Petviashvili equation with positive dispersion. Phys. Lett. A 143, 217-220.
27. Titchmarsh, E. C. 1962 Eigenfunction expansions associated with second-order differential equations. Part I, 2nd Edn. Oxford: Clarendon Press.