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The inverse Fueter mapping theorem
The bifurcation of interfacial capillary-gravity waves under O(2) symmetry
1. | School of Mathematics, Kingston University, Kingston-on-Thames, KT1 2EE01103, United Kingdom |
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. |
show all references
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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