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 and 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), 167-192. doi: 10.1017/S0305004100066500.
[2]	N. K. Bari, "A Treatise on Trigonometric Series," Pergamon Press, New York, 1964.
[3]	B. Chen and P. G. Saffman, Steady capillary-gravity waves in deep water-1.weakly nonlinear waves, Stud. App. Math., 60 (1979), 183-210.
[4]	P. Christodoulides and F. Dias, Stability of capillary-gravity interfacial waves between two bounded fluids, Phys. Fluids, 70 (1995), 3013-3027. doi: 10.1063/1.868678.
[5]	P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves, J. Fluid Mech., 265 (1994) 303-343. doi: 10.1017/S0022112094000856.
[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), 1-42. doi: 10.1016/0167-2789(82)90048-3.
[7]	M. Golubitsky and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol I," Springer, New York, 1985.
[8]	M. Golubitsky, I Stewart and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol II," Springer, New York, 1988.
[9]	M. Jones, Small amplitude capillary-gravity waves in a channel of finite depth, Glasgow Math. J., 31 (1989), 465-483.
[10]	M. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction, J. Fluid Mech., 341 (1997), 295-315. doi: 10.1017/S0022112097005624.
[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), 455-475. doi: 10.1016/S0169-5983(97)00024-5.
[12]	M. Jones, Group invariances, unfoldings and the bifurcation of capillary-gravity waves, Int. J. Bif. Chaos, 7 (1997), 1243-1266. doi: 10.1142/S021812749700100X.
[13]	M. Jones and J. F. Toland, Symmetry and the bifurcation of capillary-gravity waves, Arch. Rat. Mech. Anal., 96 (1986), 29-53. doi: 10.1007/BF00251412.
[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), 582-615. doi: 10.1007/BF01451610.
[15]	L. M. Milne-Thomson, "Theretical Hydrodynamics," 4th edition, Macmillan, New York, 1960.
[16]	H. Okamoto, On the problem of water-waves of permanent configuration, Nonlinear Analysis, Theory methods and Applications, 14 (1990), 469-481. doi: 10.1016/0362-546X(90)90035-F.
[17]	H. Okamoto, Interfacial progressive water waves- a singularity theoretic view, Tohoku Math. J., 49 (1997), 33-57.
[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), 611-640. doi: 10.2977/prims/1195165792.
[19]	J. F. Toland and M. C. W. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. R. Soc. Lond., 399 (1985), 391-417. doi: 10.1098/rspa.1985.0063.

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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), 167-192. doi: 10.1017/S0305004100066500.
[2]	N. K. Bari, "A Treatise on Trigonometric Series," Pergamon Press, New York, 1964.
[3]	B. Chen and P. G. Saffman, Steady capillary-gravity waves in deep water-1.weakly nonlinear waves, Stud. App. Math., 60 (1979), 183-210.
[4]	P. Christodoulides and F. Dias, Stability of capillary-gravity interfacial waves between two bounded fluids, Phys. Fluids, 70 (1995), 3013-3027. doi: 10.1063/1.868678.
[5]	P. Christodoulides and F. Dias, Resonant capillary-gravity interfacial waves, J. Fluid Mech., 265 (1994) 303-343. doi: 10.1017/S0022112094000856.
[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), 1-42. doi: 10.1016/0167-2789(82)90048-3.
[7]	M. Golubitsky and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol I," Springer, New York, 1985.
[8]	M. Golubitsky, I Stewart and D. Schaeffer, "Singularities and Groups in Bifurcation Theory. Vol II," Springer, New York, 1988.
[9]	M. Jones, Small amplitude capillary-gravity waves in a channel of finite depth, Glasgow Math. J., 31 (1989), 465-483.
[10]	M. Jones, Nonlinear ripples of Kelvin-Helmholtz type which arise from an interfacial mode interaction, J. Fluid Mech., 341 (1997), 295-315. doi: 10.1017/S0022112097005624.
[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), 455-475. doi: 10.1016/S0169-5983(97)00024-5.
[12]	M. Jones, Group invariances, unfoldings and the bifurcation of capillary-gravity waves, Int. J. Bif. Chaos, 7 (1997), 1243-1266. doi: 10.1142/S021812749700100X.
[13]	M. Jones and J. F. Toland, Symmetry and the bifurcation of capillary-gravity waves, Arch. Rat. Mech. Anal., 96 (1986), 29-53. doi: 10.1007/BF00251412.
[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), 582-615. doi: 10.1007/BF01451610.
[15]	L. M. Milne-Thomson, "Theretical Hydrodynamics," 4th edition, Macmillan, New York, 1960.
[16]	H. Okamoto, On the problem of water-waves of permanent configuration, Nonlinear Analysis, Theory methods and Applications, 14 (1990), 469-481. doi: 10.1016/0362-546X(90)90035-F.
[17]	H. Okamoto, Interfacial progressive water waves- a singularity theoretic view, Tohoku Math. J., 49 (1997), 33-57.
[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), 611-640. doi: 10.2977/prims/1195165792.
[19]	J. F. Toland and M. C. W. Jones, The bifurcation and secondary bifurcation of capillary-gravity waves, Proc. R. Soc. Lond., 399 (1985), 391-417. doi: 10.1098/rspa.1985.0063.

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