Abstract
Weakly non-linear behaviour of interfacial short-crested waves with current is presented in this paper. Two approaches are used to determine analytical solutions. First, a perturbation method was applied to determine the fifth-order solutions. The advantage of this method is that it allows for the determination of the harmonic resonance condition which is one of the major short-crested waves characteristics. The second method is Whitham’s Lagrangian approach. From this method, we obtained a quadratic dispersion equation. In the linear case, we have shown that there is a critical current beyond which steady wave solutions cannot exist. This critical current is associated with the emergence of instability. For the non-linear case, the critical current increases with the wave amplitude as in the two-dimensional case.
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REFERENCES
Saffman, P.G. and Yuen, H., Finite-amplitude interfacial waves in the presence of a current, J. Fluid Mech., 1982, vol. 123, pp. 459–476.
Bontozoglou, V. and Hanratty, T.J., Effects of finite depth and current velocity on large amplitude Kelvin–Helmholtz waves, J. Fluid Mech., 1988, vol. 196, pp. 187–204.
Hsu, J.R.C., Tsushiya, Y., and Silvester, R., Third-order approximation to short-crested waves, J. Fluid Mech., 1979, vol. 90, no. 1, pp. 179–196.
Roberts, A.J., Highly nonlinear short-crested water waves, J. Fluid Mech., 1983, vol. 135, pp. 301–321.
Marchant, T.R. and Roberts, A.J., Properties of short-crested waves in water of finite depth, J. Aust. Math. Soc., 1987, vol. 29, no. 1, pp. 103–125.
Tsai, C.P. and Jeng, D.S., A Fourier approximation for finite amplitude short-crested waves, J. Chinese Inst. Engineers, 1992, vol. 15, no. 6, pp. 713–721.
Wei, Z., Dalrymple, R.A., Xu, M., Garnier, R., and Derakhti, M., Short-crested waves in the surf zone, J. Geophys. Res. Oceans, 2017, vol. 122, no. 5, pp. 4143– 4162.
Kirby, J.T. and Derakhti, M., Short-crested wave breaking, European Journal of Mechanics – B/Fluids, 2019, vol. 73, pp. 100–111.
Debiane, M. and Kharif, C., Calculation of resonant short-crested waves in deep water, Phys. Fluids, 2009, vol. 21, no. 6. pp. 1–12.
Concus, P., Standing capillary-gravity waves of finite amplitude: Corrigendum, J. Fluid Mech., 1964, vol. 19, no. 2, pp. 264–266.
Plotnikov, P.I. and Toland, J.F., Nash–Moser theory for standing water waves, Arch. Rational Mech. Anal., 2001, vol. 159, no. 1, pp. 1–83.
Iooss, G., and Plotnikov, P., Asymmetrical three-dimensional travelling gravity waves, Arch. Rational Mech. Anal., 2011, vol. 200, no. 3, pp. 789–880.
Jian, Y.J., Zhu, Q.Y., Zhang, J., and Wang, Y.F. Third order approximation to capillary gravity short crested waves with uniform currents, Appl. Math. Modelling, 2009, vol. 33, no. 4, pp. 2035–2053.
Grimshaw, R. and Zhu, Y., Oblique interactions between internal solitary waves, Stud. Appl. Math., 1994, vol. 92, no. 3, pp. 249–270.
Pennell, S. and Mirie, R.M., Weak oblique collisions of interfacial solitary waves, Wave Motion, 1995, vol. 21, no. 4, pp. 385–404.
Tsuji, H. and Oikawa, M., Oblique interaction of internal solitary waves in a two-layer fluid of infinite depth, Fluid Dynam. Res, 2001, vol. 29, no. 4, pp. 251–267.
Părău, E.I., Vanden-Broeck, J.M., and Cooker, M.J., Three-dimensional gravity and gravity-capillary interfacial flows, Math. Comput. Simulation, 2007, vol. 74, no. 3, pp. 105–112.
Allalou, N., Debiane, M., and Kharif, C., Three-dimensional periodic interfacial gravity waves: Analytical and numerical results, Eur. J. Mech. B Fluids, 2011, vol. 30, no. 4, pp. 371–386.
Li, J., Liu, Z., Liao, S., and Borthwick, A.G.L., Steady-state multiple near resonances of periodic interfacial waves with rigid boundary, Phys. Fluids, 2020, vol. 32, no. 8, p. 087104.
Whitham, G.B., Linear and Non-linear Waves, Wiley, New York, 1974.
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APPENDIX
APPENDIX
Expressions for the third-order coefficients are given bellow:
$$a_{{3,1}}^{{(3)}} = \frac{1}{{{{D}_{{3,1}}}}}[( - 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} - 32{\kern 1pt} \mu {\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{3,1}}} + 16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}} - 16{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}} - 48{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu $$
$$ - \;96{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 96{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 96{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{\kern 1pt} {\kern 1pt} + 192{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 48{\kern 1pt} \omega _{0}^{2})a_{{2,0}}^{{(2)}} + ( - 8{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ - \;96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}} + 24{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 16{\kern 1pt} \mu {\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{3,1}}} - 96{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 24{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu $$
$$ + \;96{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 192{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 48{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 24{\kern 1pt} \omega _{0}^{2})a_{{2,2}}^{{(2)}} + ( - 96{\kern 1pt} \mu {\kern 1pt} p{{\omega }_{0}}{{\alpha }_{{3,1}}} + 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}U{{\alpha }_{{3,1}}}$$
$$ - \;32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - 288{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {\kern 1pt} + 288{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}})b_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (64{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{3,1}}} - 64{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}}$$
$$ + \;48{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}} - 96{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {\kern 1pt} - 48{\kern 1pt} pU\mu - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}})b_{{2,2}}^{{(2)}} + ( - 96{\kern 1pt} {{\omega }_{0}}p{{\alpha }_{{3,1}}}$$
$$ + \;32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + 288{\kern 1pt} {{p}^{2}}{{\omega }_{0}})c_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (96{\kern 1pt} {{p}^{2}}{{\omega }_{0}} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + 48{\kern 1pt} {{\omega }_{0}} - 64{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}})c_{{2,2}}^{{(2)}} + 3{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}$$
$$ + \;24{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 3{\kern 1pt} \omega _{0}^{2} - 11{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 11{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}} - 11{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}} + 3{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + 8{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ + \;\,8{\kern 1pt} \mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{3,1}}} - 6{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}} - 48{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} + 22{\kern 1pt} \mu {\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{3,1}}} + 24{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu $$
$$ - \;16{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 24{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}],$$
$$a_{{1,3}}^{{(3)}} = \frac{1}{{{{D}_{{1{\kern 1pt} {\kern 1pt} ,3}}}}}[(16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} - 64{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 32{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 32{\kern 1pt} {{p}^{4}}{{U}^{2}} - 32{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}$$
$$ + \;\,16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{1,3}}} + 96{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 48{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 16{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 48{\kern 1pt} \omega _{0}^{2} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2})a_{{0,2}}^{{(2)}} + (24{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu $$
$$ - \;32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 32{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 24{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} - 24{\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 32{\kern 1pt} {{p}^{2}}\omega _{0}^{2}$$
$$ - \;16{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 48{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 8{\kern 1pt} {{\omega }_{0}}^{2}{{\alpha }_{{1,3}}} + 64{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}})a_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} + ( - 48{\kern 1pt} pU\mu + 32{\kern 1pt} {{p}^{3}}U\mu $$
$$ - \;32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{1,3}}} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}} + 48{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} + 32{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 32{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}})b_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} $$
$$ + \;( - 32{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{1,3}}} - 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}} + 48{\kern 1pt} {{\omega }_{0}})c_{{2,2}}^{{(2)}} - 16{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 6{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 3{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{1,3}}}$$
$$ - \;3{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} + 3{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{1,3}}} + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 3{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{1,3}}} + 3{\kern 1pt} \omega _{0}^{2}$$
$$ + \;\,6{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 3{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}}],$$
$$a_{{3,3}}^{{(3)}} = - \frac{2}{{{{D}_{{3,3}}}}}[( - 8{\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 16{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}})a_{{2,2}}^{{(2)}} + (8{\kern 1pt} pU\mu - 8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}})b_{{2,2}}^{{(2)}} - {{p}^{2}}{{U}^{2}}\mu $$
$${\kern 1pt} {\kern 1pt} - \omega _{0}^{2} + 2{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - \mu {\kern 1pt} \omega _{0}^{2} + 8{{\omega }_{0}}c_{{2,2}}^{{(2)}}],$$
$$a_{{1,1}}^{{(3)}} = - (a_{{1,3}}^{{(3)}} + a_{{3,1}}^{{(3)}} + a_{{3,3}}^{{(3)}}),$$
$$b_{{1,1}}^{{(3)}} = \frac{1}{{{{D}_{{1,1}}}}}[(48{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 16{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu - 32{\kern 1pt} \omega _{0}^{2} + 16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 32{\kern 1pt} {{p}^{3}}U{{\omega }_{0}} - 32{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}}{\kern 1pt} {\kern 1pt} + 16{\kern 1pt} {{\omega }_{0}}pU)a_{{0,2}}^{{(2)}}$$
$$ + \;(48{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 16{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu - 32{\kern 1pt} \omega _{0}^{2} + 16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 32{\kern 1pt} {{p}^{3}}U{{\omega }_{0}} - 32{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} + 16{\kern 1pt} {{\omega }_{0}}pU{\kern 1pt} {\kern 1pt} )a_{{2,0}}^{{(2)}}$$
$$ + \;( - 16{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 8{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 8{\kern 1pt} {{\omega }_{0}}pU + 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} \omega _{0}^{2})a_{{2,2}}^{{(2)}} + ( - 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}} + 16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}U$$
$$ - \;16{\kern 1pt} \mu {\kern 1pt} p{{\omega }_{0}})b_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (8{\kern 1pt} pU\mu - 8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} - 16{\kern 1pt} {{\omega }_{0}})b_{{2,2}}^{{(2)}} + (32{\kern 1pt} {{p}^{2}}{{\omega }_{0}} - 16{\kern 1pt} {{\omega }_{0}}p)c_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + 16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}a_{{1,1}}^{{(3)}}$$
$$ - \;32{\kern 1pt} {{\omega }_{0}}pUa_{{1,1}}^{{(3)}} + 4{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 2{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}a_{{1,1}}^{{(3)}}{\kern 1pt} {\kern 1pt} - 8{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} + 8{\kern 1pt} {{p}^{3}}U{{\omega }_{0}} - \omega _{0}^{2} + 4{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu $$
$$ + \;4{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 32{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}pUa_{{1,1}}^{{(3)}} - {{p}^{2}}{{U}^{2}}\mu + 16{\kern 1pt} \omega _{0}^{2}a_{{1,1}}^{{(3)}}{\kern 1pt} {\kern 1pt} + {{\omega }_{0}}pU + 16{\kern 1pt} a_{{1,1}}^{{(3)}} - \mu {\kern 1pt} \omega _{0}^{2} - 16{\kern 1pt} \mu {\kern 1pt} a_{{1,1}}^{{(3)}} + 8{{\omega }_{0}}c_{{2,2}}^{{(2)}}],$$
$$c_{{1,1}}^{{(3)}} = \frac{1}{{{{D}_{{1,1}}}}}[(48{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 16{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 16{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 64{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 16{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {\kern 1pt} - 32{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}$$
$$ + \;48{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}})a_{{0,2}}^{{(2)}} + (48{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 16{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 16{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 64{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 16{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 32{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}$$
$$ + \;48{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{\kern 1pt} {\kern 1pt} )a_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (8{\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} - 8{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}})a_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (16{\kern 1pt} \mu + 16{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 16 - 16{\kern 1pt} \omega _{0}^{2}$$
$$ - \;16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2})a_{{1,1}}^{{(3)}} + ( - 16{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}U + 16{\kern 1pt} \mu {\kern 1pt} p{{\omega }_{0}} + 32{\kern 1pt} {{p}^{3}}U\mu - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}})b_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (8{\kern 1pt} pU\mu - 8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}})b_{{2,2}}^{{(2)}}$$
$$ + \;(32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}} - 32{\kern 1pt} {{p}^{3}}U\mu + 16{\kern 1pt} {{\omega }_{0}}p)c_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (8{\kern 1pt} {{\omega }_{0}} + 16{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} - 16{\kern 1pt} pU\mu )c_{{2,2}}^{{(2)}} + \omega _{0}^{2} + \mu {\kern 1pt} \omega _{0}^{2} - 4{\kern 1pt} {{p}^{2}}\omega _{0}^{2}$$
$$ - \;pU\mu {\kern 1pt} {{\omega }_{0}}{\kern 1pt} {\kern 1pt} - 4{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 12{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 16{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}}],$$
$${{\omega }_{2}} = \frac{1}{{{{D}_{{{{\omega }_{2}}}}}}}[(8{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 4{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 4{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 4{\kern 1pt} \omega _{0}^{2} - 4{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 4{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 4{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}$$
$${\kern 1pt} {\kern 1pt} - 8{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}})a_{{0,2}}^{{(2)}} + (8{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 4{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 4{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu + 4{\kern 1pt} \omega _{0}^{2} - 4{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 4{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 4{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}$$
$$ - \;8{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}})a_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + ( - 8{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 4{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 4 + 4{\kern 1pt} \omega _{0}^{2} + 4{\kern 1pt} \mu + 4{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2})a_{{1,1}}^{{(3)}} + (8{\kern 1pt} {{p}^{3}}U\mu $$
$$ - \;8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}} - 4{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}U + 4{\kern 1pt} \mu {\kern 1pt} p{{\omega }_{0}})b_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} + ( - 2{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} + 2{\kern 1pt} pU\mu )b_{{2,2}}^{{(2)}} + (4{\kern 1pt} {{\omega }_{0}}p - 8{\kern 1pt} {{p}^{2}}{{\omega }_{0}})c_{{2,0}}^{{(2)}}$$
$${\kern 1pt} {\kern 1pt} - 3{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 3{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu - 3{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 6{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 2{{\omega }_{0}}c_{{2,2}}^{{(2)}}],$$
$$b_{{1,3}}^{{(3)}} = \frac{1}{{{{\alpha }_{{1,3}}}{{D}_{{1,3}}}}}[( - 48{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 48{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} - 64{\kern 1pt} {{p}^{3}}U\omega _{0}^{2}{\kern 1pt} {\kern 1pt} - 48{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 32{\kern 1pt} {{p}^{3}}U{{\alpha }_{{1,3}}}$$
$$ + \;48{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 96{\kern 1pt} \omega _{0}^{3} - 48{\kern 1pt} pU{{\alpha }_{{1,3}}} - 16{\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{1,3}}} + 64{\kern 1pt} {{p}^{2}}\omega _{0}^{3} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}\mu {\kern 1pt} {{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} $$
$$ + \;96{\kern 1pt} pU\omega _{0}^{2} + 48{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{1,3}}} - 16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} + 16{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 16{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{1,3}}})a_{{0,2}}^{{(2)}}$$
$$ + \,(8{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} - 32{\kern 1pt} {{p}^{3}}U{{\alpha }_{{1,3}}} - 24{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 24{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 24{\kern 1pt} pU{{\alpha }_{{1,3}}} + 48{\kern 1pt} \omega _{0}^{3}$$
$$ - \;32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} - 48{\kern 1pt} pU\omega _{0}^{2} - 24{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 8{\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{1,3}}} + 64{\kern 1pt} {{p}^{3}}U\omega _{0}^{2} - 64{\kern 1pt} {{p}^{2}}\omega _{0}^{3}$$
$$ + \;24{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{1,3}}} - 24{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{1,3}}} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{1,3}}} + 8{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{1,3}}})a_{{2,2}}^{{(2)}}$$
$${\kern 1pt} {\kern 1pt} + (32{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} + 48{\kern 1pt} \omega _{0}^{2} - 32{\kern 1pt} {{p}^{2}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} - 32{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 64{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 48{\kern 1pt} {{\alpha }_{{1,3}}} + 64{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}$$
$$ + \;32{\kern 1pt} {{p}^{2}}{{\alpha }_{{1,3}}} + 48{\kern 1pt} \mu {\kern 1pt} {{\alpha }_{{1,3}}} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{1,3}}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{1,3}}})b_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} + ( - 48{\kern 1pt} \omega _{0}^{2}$$
$$ + \;32{\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 32{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{1,3}}} + 32{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} + 48{\kern 1pt} {{\omega }_{0}}pU{\kern 1pt} {\kern 1pt} - 32{\kern 1pt} {{p}^{3}}U{{\omega }_{0}} - 32{\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{1,3}}}$$
$$ + \;32{\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{1,3}}})c_{{2,2}}^{{(2)}} + 3{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} - 3{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{1,3}}} - 3{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 3{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 8{\kern 1pt} {{p}^{3}}U\omega _{0}^{2}{{\alpha }_{{1,3}}}$$
$$ - \;3{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} + 8{\kern 1pt} {{p}^{5}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} + 9{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 24{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} + 3{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}$$
$$ - \;3{\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{1,3}}} - 8{\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{1,3}}} - 24{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 3{\kern 1pt} pU{{\alpha }_{{1,3}}} - 9{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} ],$$
$$c_{{1,3}}^{{(3)}} = \frac{1}{{{{\alpha }_{{1,3}}}{{D}_{{1,3}}}}}[(64{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - 96{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 64{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3} - 16{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}}$$
$$ + \;32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} - 96{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3} + 192{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} - 32{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} - 128{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} \omega _{0}^{2} + 48{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}$$
$$ - \;48{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 16{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} + 16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}}){{a}_{{2,0,2}}} + ( - 24{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 16{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}}$$
$$ + \;8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} + 24{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 64{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3} + 48{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} + 128{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} \omega _{0}^{2} - 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}\mu {\kern 1pt} {{\alpha }_{{1,3}}}$$
$$ + \;32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 96{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} + 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3} - 8{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} + 8{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 64{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}})a_{{2,2}}^{{(2)}}$$
$$ + \;(32{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 32{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 32{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 48{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}$$
$$ - \;32{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} + 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2})b_{{2,2}}^{{(2)}} + ( - 32{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} - 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} - 48{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + 48{\kern 1pt} {{\alpha }_{{1,3}}} - 48{\kern 1pt} \mu {\kern 1pt} {{\alpha }_{{1,3}}}$$
$$ + \;32{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {\kern 1pt} + 32{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{1,3}}} + 32{\kern 1pt} {{p}^{2}}\mu {\kern 1pt} {{\alpha }_{{1,3}}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 64{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} + 96{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 32{\kern 1pt} {{p}^{2}}{{\alpha }_{{1,3}}}{\kern 1pt} )c_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} $$
$${\kern 1pt} + 3{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} - 3{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} - 3{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{1,3}}} + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 16{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}} + 6{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{1,3}}}$$
$$ - \;3{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 8{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}}{\kern 1pt} {\kern 1pt} - 3{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{1,3}}} + 8{\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{1,3}}}],$$
$$b_{{3,3}}^{{(3)}} = \frac{1}{{D{\kern 1pt} {{{\kern 1pt} }_{{3,3}}}}}[( - 8{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3} + 40{\kern 1pt} pU\omega _{0}^{2} + 24{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} + 8{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu - 8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} + 8{\kern 1pt} pU\mu - 24{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}$$
$${\kern 1pt} {\kern 1pt} - 8{\kern 1pt} pU + 8{\kern 1pt} {{\omega }_{0}} - 40{\kern 1pt} \omega _{0}^{3})a_{{2,2}}^{{(2)}} + ( - 64{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} - 16 + 32{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + 32{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} + 16{\kern 1pt} \mu + 48{\kern 1pt} \omega _{0}^{2})b_{{2,2}}^{{(2)}}$$
$$ + \;( - 16{\kern 1pt} \omega _{0}^{2} + 16{\kern 1pt} {{\omega }_{0}}pU)c_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} - {{p}^{3}}{{U}^{3}}\mu - 3{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} + pU + 3{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - pU\mu - {{\omega }_{0}} + \mu {\kern 1pt} \omega _{0}^{3} - pU\omega _{0}^{2} + \mu {\kern 1pt} {{\omega }_{0}} + \omega _{0}^{3}],$$
$$c_{{3,3}}^{{(3)}} = - \frac{1}{{{{D}_{{3{\kern 1pt} {\kern 1pt} ,3}}}}}[(8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}} + 40{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3} + 8{\kern 1pt} \omega _{0}^{3} - 80{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} + 40{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - 8{\kern 1pt} {{\omega }_{0}})a_{{2,2}}^{{(2)}} + ( - 16{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}$$
$$ + \;16{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}})b_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} + (48{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + 16{\kern 1pt} \mu + 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} - 16 + 32{\kern 1pt} \omega _{0}^{2} - 96{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}})c_{{2,2}}^{{(2)}}$$
$$ + \;\mu {\kern 1pt} {{\omega }_{0}} + {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - {{\omega }_{0}} - 2{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} + \omega _{0}^{3} + \mu {\kern 1pt} \omega _{0}^{3}],$$
$$b_{{3,1}}^{{(3)}} = \frac{1}{{{{\alpha }_{{3,1}}}{{D}_{{3,1}}}}}[ - 8{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}} + 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - \mu {\kern 1pt} pU{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - {{\omega }_{0}}{{\alpha }_{{3,1}}} + \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}} - 8{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}}$$
$$ + \;(192{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}} + 192{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}} - 96{\kern 1pt} \mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{3,1}}} + 16{\kern 1pt} \mu {\kern 1pt} {{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 192{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}} + 288{\kern 1pt} {{p}^{2}}\omega _{0}^{2}$$
$$ + \,32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}} - 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}} - 384{\kern 1pt} \mu {\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 32{\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}} + 144{\kern 1pt} \omega _{0}^{2} - 16{\kern 1pt} {{\alpha }_{{3,1}}})b_{{2,2}}^{{(2)}}$$
$$ + \;(864{\kern 1pt} {{p}^{3}}U{{\omega }_{0}} + 288{\kern 1pt} \omega _{0}^{2}p{{\alpha }_{{3,1}}} - 96{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}} - 864{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{\kern 1pt} {\kern 1pt} - 288{\kern 1pt} {{p}^{2}}U{{\omega }_{0}}{{\alpha }_{{3,1}}} + 96{\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}})c_{{2,0}}^{{(2)}}{\kern 1pt} {\kern 1pt} $$
$$ + \,33{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} + (288{\kern 1pt} {{p}^{3}}{{U}^{2}}\mu {\kern 1pt} {{\alpha }_{{3,1}}} + 192{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}} + 864{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}} - 96{\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} $$
$$ - \;576{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}U{{\alpha }_{{3,1}}} - 96{\kern 1pt} \mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{3,1}}} + 288{\kern 1pt} \mu {\kern 1pt} p\omega _{0}^{2}{{\alpha }_{{3,1}}} - 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} )b_{{2,0}}^{{(2)}} + (288{\kern 1pt} {{p}^{3}}U{{\omega }_{0}}$$
$$ - \;192{\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{3,1}}} - 96{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}} + 96{\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 192{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}} - 288{\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 144{\kern 1pt} {{\omega }_{0}}pU$$
$$ - \;144{\kern 1pt} \omega _{0}^{2})c_{{2,2}}^{{(2)}} + ( - 24{\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} - 32{\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + 24{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 8{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{3,1}}}$$
$$ - \;24{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} + 576{\kern 1pt} {{p}^{3}}U\omega _{0}^{2} + 8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 576{\kern 1pt} {{p}^{2}}\omega _{0}^{3} + 72{\kern 1pt} \mu {\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ + \;24{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} + 8{\kern 1pt} pU{{\alpha }_{{3,1}}} - 72{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 144{\kern 1pt} pU\omega _{0}^{2} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - 8{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}}$$
$$ + \;144{\kern 1pt} \omega _{0}^{3})a_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} + ( - 32{\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}} - 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} - 16{\kern 1pt} pU{{\alpha }_{{3,1}}} + 288{\kern 1pt} pU\omega _{0}^{2} + 16{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} $$
$$ + \;16{\kern 1pt} \mu {\kern 1pt} pU{{\alpha }_{{3,1}}} + 48{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{3,1}}} - 48{\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} + 48{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} - 144{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}}$$
$$ + \;144{\kern 1pt} \mu {\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} + 32{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}} + 576{\kern 1pt} {{p}^{3}}U\omega _{0}^{2} - 16{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - 288{\kern 1pt} \omega _{0}^{3}$$
$$ - \;576{\kern 1pt} {{p}^{2}}\omega _{0}^{3})a_{{2,0}}^{{(2)}} + 99{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}} - 99{\kern 1pt} \mu {\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} - 33{\kern 1pt} {{p}^{3}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 72{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U\omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ - \;33{\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} - 24{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{3,1}}} + 24{\kern 1pt} {{p}^{5}}{{U}^{3}}\mu {\kern 1pt} {{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 24{\kern 1pt} {{p}^{3}}U\omega _{0}^{2}{{\alpha }_{{3,1}}} - 72{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{3,1}}}$$
$$ - \;24{\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{3,1}}} + 8{\kern 1pt} {{p}^{3}}U{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 33{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} + pU{{\alpha }_{{3,1}}}],$$
$$c_{{3,1}}^{{(3)}} = \frac{1}{{{{\alpha }_{{3,1}}}{{D}_{{3,1}}}{\kern 1pt} {\kern 1pt} }}[( - 576{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - 16{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}} - 288{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3} + 1152{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} \omega _{0}^{2}{\kern 1pt} + 48{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}}$$
$$ - \;576{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + 576{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2} + 48{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 288{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - 96{\kern 1pt} \mu {\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ - \;32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - 48{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 16{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}})a_{{2,0}}^{{((2)}} + (24{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}} - 8{\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}} - 576{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}}$$
$${\kern 1pt} {\kern 1pt} + 24{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} + 32{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - 48{\kern 1pt} \mu {\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} + 1152{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} \omega _{0}^{2}{\kern 1pt} {\kern 1pt} + 144{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3} - 24{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}}$$
$$ - \;32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + 144{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {{\omega }_{0}} - 288{\kern 1pt} pU\mu {\kern 1pt} \omega _{0}^{2}{\kern 1pt} {\kern 1pt} + 8{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}} - 576{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3})a_{{2,2}}^{{(2)}}{\kern 1pt} {\kern 1pt} + ( - 288{\kern 1pt} \mu {\kern 1pt} p\omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ - \;96{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}} + 864{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 864{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} + 288{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}U{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}})b_{{2,0}}^{{(2)}}$$
$$ + \;(192{\kern 1pt} \mu {\kern 1pt} pU{{\omega }_{0}}{{\alpha }_{{3,1}}} + 288{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 288{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 192{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}} - 144{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}{\kern 1pt} {\kern 1pt} + 96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}}$$
$$ - \;96{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U{{\omega }_{0}}{{\alpha }_{{3,1}}} + 144{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2}){{b}_{{2,2,2}}}{\kern 1pt} {\kern 1pt} + ( - 864{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu - 864{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} + 1728{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}} - 288{\kern 1pt} \omega _{0}^{2}p{{\alpha }_{{3,1}}}$$
$$ - \;96{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 96{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}} + 96{\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}})c_{{2,0}}^{{(2)}} + (32{\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}} - 192{\kern 1pt} \omega _{0}^{2}{{\alpha }_{{3,1}}} - 16{\kern 1pt} \mu {\kern 1pt} {{\alpha }_{{3,1}}} - 32{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\alpha }_{{3,1}}}$$
$$ - \;288{\kern 1pt} {{p}^{4}}{{U}^{2}}\mu {\kern 1pt} {\kern 1pt} + 96{\kern 1pt} {{p}^{2}}\omega _{0}^{2}{{\alpha }_{{3,1}}} + 288{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 16{\kern 1pt} {{\alpha }_{{3,1}}} - 144{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} - 288{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{2} - 144{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu {\kern 1pt} {\kern 1pt} $$
$$ + \;576{\kern 1pt} {{p}^{3}}U\mu {\kern 1pt} {{\omega }_{0}})c_{{2,2}}^{{(2)}} - 48{\kern 1pt} \mu {\kern 1pt} {{p}^{3}}U\omega _{0}^{2}{{\alpha }_{{3,1}}} + 24{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 66{\kern 1pt} \mu {\kern 1pt} pU\omega _{0}^{2}{{\alpha }_{{3,1}}} - 8{\kern 1pt} \mu {\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} + {{\omega }_{0}}{{\alpha }_{{3,1}}}$$
$$ - \;33{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} - 33{\kern 1pt} \omega _{0}^{3}{{\alpha }_{{3,1}}} + 24{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{4}}{{U}^{2}}{{\alpha }_{{3,1}}} - \mu {\kern 1pt} {{\omega }_{0}}{{\alpha }_{{3,1}}} + 24{\kern 1pt} {{p}^{2}}\omega _{0}^{3}{{\alpha }_{{3,1}}}{\kern 1pt} {\kern 1pt} + 8{\kern 1pt} {{p}^{2}}{{\omega }_{0}}{{\alpha }_{{3,1}}} - 33{\kern 1pt} {{\omega }_{0}}\mu {\kern 1pt} {{p}^{2}}{{U}^{2}}{{\alpha }_{{3,1}}}],$$
$${{D}_{{3,1}}} = 32({\kern 1pt} 9{\kern 1pt} \omega _{0}^{2} - {{\alpha }_{{3,1}}} + 9{\kern 1pt} \mu {\kern 1pt} \omega _{0}^{2} - 18{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + 9{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + \mu {\kern 1pt} {{\alpha }_{{3,1}}}),$$
$${{D}_{{3,3}}} = 32({\kern 1pt} - 1 + \mu + 3{\kern 1pt} {{\omega }_{0}}^{2} + 3{\kern 1pt} {{p}^{2}}{{U}^{2}}\mu + 3{\kern 1pt} \mu {\kern 1pt} {{\omega }_{0}}^{2} - 6{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}}),$$
$${{D}_{{1,3}}} = 32({\kern 1pt} {{p}^{2}}{{U}^{2}}\mu - 2{\kern 1pt} pU\mu {\kern 1pt} {{\omega }_{0}} + \mu {\kern 1pt} \omega _{0}^{2} - {{\alpha }_{{1,3}}} + \mu {\kern 1pt} {{\alpha }_{{1,3}}} + \omega _{0}^{2}),$$
$${{D}_{{1,1}}} = 32({\kern 1pt} pU\mu - {{\omega }_{0}} - \mu {\kern 1pt} {{\omega }_{0}}),$$
$${{D}_{{{{\omega }_{2}}}}} = 8({\kern 1pt} pU\mu - {{\omega }_{0}} - \mu {\kern 1pt} {{\omega }_{0}}).$$
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Salmi, S., Allalou, N. & Debiane, M. Weakly Nonlinear Gravity Three-Dimensional Unbounded Interfacial Waves: Perturbation Method and Variational Formulation. Fluid Dyn 56 (Suppl 1), S53–S69 (2021). https://doi.org/10.1134/S0015462822010098
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DOI: https://doi.org/10.1134/S0015462822010098