Abstract
A nonlinear Schrödinger equation (NSE) describing packets of weakly nonlinear waves in an inhomogeneously vortical infinitely deep fluid has been derived. The vorticity is assumed to be an arbitrary function of Lagrangian coordinates and quadratic in the small parameter proportional to the wave steepness. It is shown that the modulational instability criteria for the weakly vortical waves and potential Stokes waves on deep water coincide. The effect of vorticity manifests itself in a shift of the wavenumber of high-frequency filling. A special case of Gerstner waves with a zero coefficient at the nonlinear term in the NSE is noted.
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References
V. E. Zakharov, “Stability of periodic waves of finite amplitude on a surface of deep fluid. J Appl Mech Tech Phys 2: 190–194 (1968).
H. Hasimoto and H. Ono, “Nonlinear modulation of gravity waves,” J. Phys. Soc. Jpn. 33, 805–811 (1972).
A. Davey, “The propagation of a weak nonlinear wave,” J. Fluid Mech. 53, 769–781 (1972).
H. C. Yuen and B. M. Lake, “Nonlinear deep water waves: Theory and experiment,” Phys. Fluids 18, 956–960 (1975).
R. S. Johnson, “On the modulation of water waves on shear flows,” Proc. R. Soc. London, Ser. A 347, 537–546 (1976).
A. I. Baumstein, “Modulation of gravity waves with shear in water,” Stud. Appl. Math. 100, 365–390 (1998).
R. Thomas, C. Kharif, and M. Manna, “A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity,” Phys. Fluids 24, 127102 (2012).
K. B. Hjelmervik and K. Trulsen, “Freak wave statistics on collinear currents,” J. Fluid Mech. 637, 267–284 (2009).
A. A. Abrashkin, “Gravity surface waves of the envelope in a vortical fluid,” Izv. Akad. Nauk: Fiz. Atmos. Okeana 27 (6), 633–637 (1991).
A. A. Abrashkin and E. I. Yakubovich, Vortex Dynamics in Lagrangian Description (Fizmatlit, Moscow, 2006) [in Russian].
L. N. Sretenskii, Theory of Wave Motions of Fluids (Nauka, Moscow, 1977) [in Russian].
D. H. Peregrine, “Water waves, nonlinear Schrödinger equations and their solutions,” J. Aust. Math. Soc., Ser. B 25, 16–43 (1983).
N. N. Akhmediev, V. M. Eleonskii, and N. E. Kulagin, “Generation of periodic trains of picosecond pulses in an optical fiber: Exact solutions,” Sov. Phys. JETP 62 (5), 894–899 (1985).
E. A. Kuznetsov, “Solitons in a parametric instable plasma,” Sov. Phys. Dokl. 22, 507–508 (1977).
A. Slunyaev, I. Didenkulova, and E. Pelinovsky, “Rogue waves,” Contemp. Phys. 52, 571–590 (2011).
A. A. Gelash and V. E. Zakharov, “Superregular solitonic solutions: A novel scenario for the nonlinear stage of modulation instability,” Nonlinearity 27, R1–R39 (2014).
V. P. Ruban, “On the nonlinear Schrödinger equation for waves on a nonuniform current,” JETP Lett. 95 (9), 486–491 (2012).
J. He and Y. Li, “Designable integrability of the variable coefficient nonlinear Schrödinger equations,” Stud. Appl. Math. 126, 1–15 (2010).
H.-H. Chen and C.-S. Liu, “Solitons in nonuniform media,” Phys. Rev. Lett. 37, 693–697 (1976).
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Original Russian Text © A.A. Abrashkin, E.N. Pelinovsky, 2018, published in Izvestiya Rossiiskoi Akademii Nauk, Fizika Atmosfery i Okeana, 2018, Vol. 54, No. 1, pp. 112–117.
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Abrashkin, A.A., Pelinovsky, E.N. Dynamics of a Wave Packet on the Surface of an Inhomogeneously Vortical Fluid (Lagrangian Description). Izv. Atmos. Ocean. Phys. 54, 101–105 (2018). https://doi.org/10.1134/S0001433818010036
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DOI: https://doi.org/10.1134/S0001433818010036