Skip to main content Accessibility help
×
Home

Water waves for small surface tension: an approach via normal form

  • Gérard Iooss (a1) and Klaus Kirchgässner (a2)

Synopsis

In this paper we determine the possible crest-forms of permanent waves of small amplitude which exist on the free surface of a two-dimensional fluid layer under the influence of gravity and surface tension when the Froude number is close to 1. The Bond number b, measuring surface tension, is assumed to satisfy b < ⅓. We find one-parameter families of periodic waves of two different types, quasiperiodic waves and solitary waves with oscillations at infinity. The existence of true solitary waves is established for a sequence of systems approximating the full Euler equations in every algebraic order of − 1.

Copyright

References

Hide All
1Amick, C. J. and Kirchgässner, K.. Solitary water-waves in the presence of surface tension. In Dynamical Problems in Continuum Physics, I.M.A. Volumes in Mathematics and Its Applications 4 (Berlin: Springer, 1986).
2Amick, C. J. and Kirchgässner, K.. A theory of solitary water-waves in the presence of surface tension. Arch. Rational Mech. Anal. 105 (1989), 149.
3Amick, C. J. and McLeod, J. B.. A singular perturbation problem in water waves (manuscript).
4Amick, C. J. and Toland, J. F.. Trajectories homoclinic to period orbits in four dimensions (manuscript).
5Beale, J. T.. Exact solitary water waves with capillary ripples at infinity. Comm. Pure Appl. Math. 44 (1991), 211257.
6Coullet, P. and Spiegel, M. E.. Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43 (1983), 774819.
7Dias, F., Iooss, G. and Vanden-Broeck, J. M.. Capillary-gravity solitary waves with damped oscillations (manuscript).
8Eckhaus, W.. Singular perturbations of homoclinic orbits in ℝ4 (University of Utrecht, Preprint, Nr. 642, 1991).
9Elphick, C., Tirapegui, E., Brachet, M. E., Coullet, P. and Iooss, G.. A simple global characterization for normal forms of singular vector fields. Physica D 29 (1987), 95127.
10Hamilton, R. S.. The inverse function theorem of Nash-Moser. Bull. Amer. Math. Soc. 7 (1982), 65222.
11Hammersley, J. M. and Mazzarino, G.. Computational aspects of some autonomous differential equations. Proc. Roy. Soc. London Ser. A 424 (1989), 1937.
12Hunter, J. K. and Scheurle, J.. Existence of perturbed solitary wave solutions to a model equation for water waves. Physica D 32 (1988), 253268.
13Iooss, G. and Adelmeyer, M.. Topics in bifurcation theory and applications (manuscript).
14Iooss, G. and Kirchgässner, K.. Bifurcations d'ondes solitaires en présense d'une faible tension superficielle. Note C.R. Acad. Sci. Paris 311 1 (1990), 265268.
15Iooss, G. and Los, J.. Bifurcation of spatially quasi-periodic solutions in hydrodynamic stability problems. Nonlinearity 3 (1990), 851871.
16Iooss, G. and Peroueme, M. C.. Perturbed homoclinic solution in reversible 1:1 resonance vector fields. J. Differential Equations (to appear).
17Kato, T.. Perturbation theory for linear operators, Grundlehren der mathematischen Wissenschaften (New York: Springer, 1966).
18Kirchgässner, K.. Wave solutions of reversible systems and applications. J. Differential Equations 45 (1982), 113127.
19Kirchgässner, K.. Nonlinearly resonant surface waves and homoclinic bifurcation. Adv. in Appl. Mech. 26 (1988), 135181.
20Mielke, A.. Reduction of quasilinear elliptic equations in cylindrical domains with applications. Math. Methods Appl. Sci. 10 (1988), 5166.
21Sun, S. M.. Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3. J. Math. Anal. Appl. 156 (1991), 471504.
22Whitham, G. B.. Linear and nonlinear waves (New York: J. Wiley, 1974).
23Zeidler, E.. Existenzbeweis für cnoidal waves unter Berücksichtigung der Oberflächenspannung. Arch. Rational Mech. Anal. 41 (1971), 81107.

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed