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On the homotopy multiple-variable method and its applications in the interactions of nonlinear gravity waves. (English) Zbl 1221.76046
Summary: The basic ideas of a homotopy-based multiple-variable method is proposed and applied to investigate the nonlinear interactions of periodic traveling waves. Mathematically, this method does not depend upon any small physical parameters at all and thus is more general than the traditional multiple-scale perturbation techniques. Physically, it is found that, for a fully developed wave system, the amplitudes of all wave components are finite even if the wave resonance condition given by {\it O. M. Phillips} [J. Fluid Mech. 9, 193--217 (1960; Zbl 0094.41101)] is exactly satisfied. Besides, it is revealed that there exist multiple resonant waves, and that the amplitudes of resonant wave may be much smaller than those of primary waves so that the resonant waves sometimes contain rather small part of wave energy. Furthermore, a wave resonance condition for arbitrary numbers of traveling waves with large wave amplitudes is given, which logically contains Phillips’ four-wave resonance condition but opens a way to investigate the strongly nonlinear interaction of more than four traveling waves with large amplitudes. This work also illustrates that the homotopy multiple-variable method is helpful to gain solutions with important physical meanings of nonlinear problems, if the multiple-variables are properly defined with clear physical meanings.

76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
65N99Numerical methods for BVP of PDE
76M25Other numerical methods (fluid mechanics)
Full Text: DOI arXiv
[1] Phillips, O. M.: On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions, J fluid mech 9, 193-217 (1960) · Zbl 0094.41101 · doi:10.1017/S0022112060001043
[2] Longuet-Higgins, M. S.: Resonant interactions between two trains of gravity waves, J fluid mech 12, 321-332 (1962) · Zbl 0105.20302 · doi:10.1017/S0022112062000233
[3] Longuet-Higgins, M. S.; Smith, N. D.: An experiment on third order resonant wave interactions, J fluid mech 25, 417-435 (1966)
[4] Mcgoldrick, L. F.; Phillips, O. M.; Huang, N.; Hodgson, T.: Measurements on resonant wave interactions, J fluid mech 25, 437-456 (1966)
[5] Benney, D. T.: Non-linear gravity wave interactions, J fluid mech 14, 577-584 (1962) · Zbl 0117.43605 · doi:10.1017/S0022112062001469
[6] Bretherton, P.: Resonant interactions between waves: the case of discrete oscillations, J fluid mech 20, 457-479 (1964)
[7] Phillips, O. M.: Wave interactions- the evolution of an idea, J fluid mech 106, 215-227 (1981) · Zbl 0467.76020 · doi:10.1017/S0022112081001572
[8] Liao, S. J.: Beyond perturbation: introduction to the homotopy analysis method, (2003)
[9] Liao, S. J.: An explicit, totally analytic approximation of Blasius viscous flow problems, Int J nonlinear mech 34, No. 4, 759-778 (1999) · Zbl 05137896
[10] Liao, S. J.: A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J fluid mech 385, 101-128 (1999) · Zbl 0931.76017 · doi:10.1017/S0022112099004292
[11] Liao, S. J.; Campo, A.: Analytic solutions of the temperature distribution in Blasius viscous flow problems, J fluid mech 453, 411-425 (2002) · Zbl 1007.76014 · doi:10.1017/S0022112001007169
[12] Liao, S. J.: On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J fluid mech 488, 189-212 (2003) · Zbl 1063.76671 · doi:10.1017/S0022112003004865
[13] Liao, S. J.: Series solutions of unsteady boundary-layer flows over a stretching flat plate, Stud appl math 117, No. 3, 2529-2539 (2006) · Zbl 1145.76352 · doi:10.1111/j.1467-9590.2006.00354.x
[14] Liao, S. J.; Tan, Y.: A general approach to obtain series solutions of nonlinear differential equations, Stud appl math 119, 297-355 (2007)
[15] Xu, H.; Lin, Z. L.; Liao, S. J.; Wu, J. Z.; Majdalani, J.: Homotopy-based solutions of the Navier -- Stokes equations for a porous channel with orthogonally moving walls, Phys fluids 22 (2010) · Zbl 1190.76132 · doi:10.1063/1.3392770
[16] Li YJ, Nohara BT, Liao SJ. Series solutions of coupled Van der Pol equation by means of homotopy analysis method. J Math Phys.
[17] Liao, S. J.: An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun nonlinear sci numer simul 15, 2003-2016 (2010) · Zbl 1222.65088 · doi:10.1016/j.cnsns.2009.09.002
[18] Annenkov, S. Y.; Shrira, V. I.: Role of non-resonant interactions in the evolution of nonlinear random water wave fields, J fluid mech 561, 181-207 (2006) · Zbl 1157.76309 · doi:10.1017/S0022112006000632
[19] Kharif, C.; Pelinovsky, E.: Physical mechanisms of the rogue wave phenomenon, Eur J mech B-fluids 22, 603-634 (2003) · Zbl 1058.76017 · doi:10.1016/j.euromechflu.2003.09.002
[20] Gibbs, R. H.; Taylor, P. H.: Formation of walls of water in ’fully’ nonlinear simulations, Appl ocean res 27, 142-257 (2005)
[21] Adcock, T. A. A.; Taylor, P. H.: Focusing of unidirectional wave groups on deep water: an approximate nonlinear Schrödinger equation-based model, Proc R soc A 465, 3083-3102 (2009) · Zbl 1181.35182 · doi:10.1098/rspa.2009.0224