The bifurcation of interfacial capillary-gravity waves under O(2) symmetry

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July 2011, 10(4): 1183-1204. doi: 10.3934/cpaa.2011.10.1183

The bifurcation of interfacial capillary-gravity waves under O(2) symmetry

1.	School of Mathematics, Kingston University, Kingston-on-Thames, KT1 2EE01103, United Kingdom

Received March 2010 Revised November 2010 Published April 2011

Abstract
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The evolution of an interface between two fluids of different densities is considered. The particular case under examination is when the motion is due to an interaction between the Mth and Nth harmonics of the fundamental mode. By means of a hodograph transformation the problem is cast as an operator equation between two suitable function spaces. Classical techniques are used to reduce the problem to a finite system of algebraic equations. Solutions are found which exhibit a rich variety of behaviour including primary, secondary and multiple bifurcation.

Keywords: Bifurcation, fluid-fluid interface, group invariance., Lyapunov-Schmidt.

Mathematics Subject Classification: Primary: 37G40, 76B15; Secondary: 34G5.

Citation: Mark Jones. The bifurcation of interfacial capillary-gravity waves under O(2) symmetry. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1183-1204. doi: 10.3934/cpaa.2011.10.1183

References:

[1]	D. Armbruster and G. Dangelmayr, Coupled stationary bifurcations in nonflux boundary value problems,, Math. Proc. Camb. Phil. Soc., 101 (1987), 167. doi: 10.1017/S0305004100066500. Google Scholar
[2]	N. K. Bari, "A Treatise on Trigonometric Series,", Pergamon Press, (1964). Google Scholar
[3]	B. Chen and P. G. Saffman, Steady capillary-gravity waves in deep water-1.weakly nonlinear waves,, Stud. App. Math., 60 (1979), 183. Google Scholar
[4]	P. Christodoulides and F. Dias, Stability of capillary-gravity interfacial waves between two bounded fluids,, Phys. Fluids, 70 (1995), 3013. doi: 10.1063/1.868678. Google Scholar
[5]	P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves,, J. Fluid Mech., 265 (1994), 303. doi: 10.1017/S0022112094000856. Google Scholar
[6]	H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems,, Physica D, 5 (1982), 1. doi: 10.1016/0167-2789(82)90048-3. Google Scholar
[7]	M. Golubitsky and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol I,", Springer, (1985). Google Scholar
[8]	M. Golubitsky, I Stewart and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol II,", Springer, (1988). Google Scholar
[9]	M. Jones, Small amplitude capillary-gravity waves in a channel of finite depth,, Glasgow Math. J., 31 (1989), 465. Google Scholar
[10]	M. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction,, J. Fluid Mech., 341 (1997), 295. doi: 10.1017/S0022112097005624. Google Scholar
[11]	M. Jones, A system of equations which predicts the nonlinear evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption,, Fluid Dyn. Res., 21 (1997), 455. doi: 10.1016/S0169-5983(97)00024-5. Google Scholar
[12]	M. Jones, Group invariances, unfoldings and the bifurcation of capillary-gravity waves,, Int. J. Bif. Chaos, 7 (1997), 1243. doi: 10.1142/S021812749700100X. Google Scholar
[13]	M. Jones and J. F. Toland, Symmetry and the bifurcation of capillary-gravity waves,, Arch. Rat. Mech. Anal., 96 (1986), 29. doi: 10.1007/BF00251412. Google Scholar
[14]	N. Kotchine, Determination rigoureuse des ondes permanentes d'ampleur finie a la surface de seperation deux liquides de profondeur finie,, Math. Ann., 98 (1928), 582. doi: 10.1007/BF01451610. Google Scholar
[15]	L. M. Milne-Thomson, "Theretical Hydrodynamics," 4th, edition, (1960). Google Scholar
[16]	H. Okamoto, On the problem of water-waves of permanent configuration,, Nonlinear Analysis, 14 (1990), 469. doi: 10.1016/0362-546X(90)90035-F. Google Scholar
[17]	H. Okamoto, Interfacial progressive water waves- a singularity theoretic view,, Tohoku Math. J., 49 (1997), 33. Google Scholar
[18]	H. Okamoto and M. Shoji, Remarks on the bifurcation of two dimensional capillary-gravity waves of finite depth,, Publ. Res. Inst. Math. Sci., 30 (1994), 611. doi: 10.2977/prims/1195165792. Google Scholar
[19]	J. F. Toland and M. C. W. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. R. Soc. Lond., 399 (1985), 391. doi: 10.1098/rspa.1985.0063. Google Scholar

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References:

[1]	D. Armbruster and G. Dangelmayr, Coupled stationary bifurcations in nonflux boundary value problems,, Math. Proc. Camb. Phil. Soc., 101 (1987), 167. doi: 10.1017/S0305004100066500. Google Scholar
[2]	N. K. Bari, "A Treatise on Trigonometric Series,", Pergamon Press, (1964). Google Scholar
[3]	B. Chen and P. G. Saffman, Steady capillary-gravity waves in deep water-1.weakly nonlinear waves,, Stud. App. Math., 60 (1979), 183. Google Scholar
[4]	P. Christodoulides and F. Dias, Stability of capillary-gravity interfacial waves between two bounded fluids,, Phys. Fluids, 70 (1995), 3013. doi: 10.1063/1.868678. Google Scholar
[5]	P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves,, J. Fluid Mech., 265 (1994), 303. doi: 10.1017/S0022112094000856. Google Scholar
[6]	H. Fujii, M. Mimura and Y. Nishiura, A picture of the global bifurcation diagram in ecological interacting and diffusing systems,, Physica D, 5 (1982), 1. doi: 10.1016/0167-2789(82)90048-3. Google Scholar
[7]	M. Golubitsky and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol I,", Springer, (1985). Google Scholar
[8]	M. Golubitsky, I Stewart and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol II,", Springer, (1988). Google Scholar
[9]	M. Jones, Small amplitude capillary-gravity waves in a channel of finite depth,, Glasgow Math. J., 31 (1989), 465. Google Scholar
[10]	M. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction,, J. Fluid Mech., 341 (1997), 295. doi: 10.1017/S0022112097005624. Google Scholar
[11]	M. Jones, A system of equations which predicts the nonlinear evolution of a wave packet due to a fluid-fluid interaction under a narrow bandwidth assumption,, Fluid Dyn. Res., 21 (1997), 455. doi: 10.1016/S0169-5983(97)00024-5. Google Scholar
[12]	M. Jones, Group invariances, unfoldings and the bifurcation of capillary-gravity waves,, Int. J. Bif. Chaos, 7 (1997), 1243. doi: 10.1142/S021812749700100X. Google Scholar
[13]	M. Jones and J. F. Toland, Symmetry and the bifurcation of capillary-gravity waves,, Arch. Rat. Mech. Anal., 96 (1986), 29. doi: 10.1007/BF00251412. Google Scholar
[14]	N. Kotchine, Determination rigoureuse des ondes permanentes d'ampleur finie a la surface de seperation deux liquides de profondeur finie,, Math. Ann., 98 (1928), 582. doi: 10.1007/BF01451610. Google Scholar
[15]	L. M. Milne-Thomson, "Theretical Hydrodynamics," 4th, edition, (1960). Google Scholar
[16]	H. Okamoto, On the problem of water-waves of permanent configuration,, Nonlinear Analysis, 14 (1990), 469. doi: 10.1016/0362-546X(90)90035-F. Google Scholar
[17]	H. Okamoto, Interfacial progressive water waves- a singularity theoretic view,, Tohoku Math. J., 49 (1997), 33. Google Scholar
[18]	H. Okamoto and M. Shoji, Remarks on the bifurcation of two dimensional capillary-gravity waves of finite depth,, Publ. Res. Inst. Math. Sci., 30 (1994), 611. doi: 10.2977/prims/1195165792. Google Scholar
[19]	J. F. Toland and M. C. W. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves,, Proc. R. Soc. Lond., 399 (1985), 391. doi: 10.1098/rspa.1985.0063. Google Scholar

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