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FOURTH-ORDER NONLINEAR SCHRÖDINGER EQUATION FOR APPLICATION TO CAPILLARY-GRAVITY WAVES IN DEEP WATER ON FLOWS OF BULK VORTICITY

Part of: Incompressible inviscid fluids

Published online by Cambridge University Press:  18 November 2025

DEBRAJ GIRI ,
TANMOY PAL Open the ORCID record for TANMOY PAL [Opens in a new window]  and
ASOKE KUMAR DHAR Open the ORCID record for ASOKE KUMAR DHAR [Opens in a new window]
DEBRAJ GIRI
Affiliation:
Mathematics, Indian Institute of Engineering Science and Technology , India; e-mail: debrajgiri183@gmail.com
TANMOY PAL
Affiliation:
Mathematics, Swami Vivekananda University, Barrackpore, West Bengal , India; e-mail: tpal2996@gmail.com
ASOKE KUMAR DHAR*
Affiliation:
Mathematics, Indian Institute of Engineering Science and Technology , India; e-mail: debrajgiri183@gmail.com
*
e-mail: asoke.dhar@gmail.com
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Abstract

The modulational instability of weakly nonlinear capillary-gravity waves (CGWs) on the surface of infinitely deep water with uniform vorticity background shear is examined. Assuming a narrow band of waves, the fourth-order nonlinear Schrödinger equation (NSE) is derived from Zakharov’s integral equation (ZIE). The analysis is restricted to one horizontal dimension, parallel to the direction along the wave propagation to take advantage of a formulation using potential flow theory. It is to be noted that the dominant new effect introduced to the fourth order is the wave-induced mean flow response. The key point of this paper is that the present fourth-order analysis shows considerable deviation in the stability properties of CGWs from the third-order analysis and gives better results consistent with the exact results. It is found that the growth rate of instability increases for negative vorticity and decreases for positive vorticity, and the effect of capillarity is to reduce the growth rate of instability. Additionally, the effect of vorticity on the Peregrine breather, which can be considered as a prototype for freak waves, is investigated.

Keywords

fourth-order nonlinear Schrödinger equationbulk vorticitycapillary-gravity wavesstability analysis

MSC classification

Primary: 76B07: Free-surface potential flows
Secondary: 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction 76B45: Capillarity (surface tension)

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 67 , 2025 , e37
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of the Australian Mathematical Publishing Association Inc.

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References

Alber, I. E., “The effects of randomness on the stability of two-dimensional surface wavetrains”, Proc. Roy. Soc. London Sect. A 363(1715) (1978) 525–546; doi:10.1098/rspa.1978.0181.Google Scholar
Alber, I. E. and Saffman, P. G., “Stability of random nonlinear deep water waves with finite bandwidth spectra”, TRW (Defense and Space Systems Group, Redondo Beach, CA, 1978); https://www.mn.uio.no/math/personer/vit/karstent/alber-1978-stability.pdf.Google Scholar
Benjamin, T. B. and Feir, J. E., “The disintegration of wave trains on deep water Part 1. Theory”, J. Fluid Mech. 27 (1967) 417–430; doi:10.1017/S002211206700045X.CrossRefGoogle Scholar
Caponi, E. A., Yuen, H. C., Milinazzo, F. A. and Saffman, P. G., “Water-wave instability induced by a drift layer”, J. Fluid Mech. 222 (1991) 207–213; doi:10.1017/S0022112091001064.CrossRefGoogle Scholar
Chabchoub, A., Akhmediev, N. and Hoffmann, N. P., “Experimental study of spatiotemporally localized surface gravity water waves”, Phys. Rev. E 86(3) (2012) article ID: 016311; doi:10.1103/PhysRevE.86.016311.CrossRefGoogle ScholarPubMed
Crapper, G. D., “An exact solution for progressive capillary waves of arbitrary amplitude”, J. Fluid Mech. 2 (1957) 532–540; doi:10.1017/S0022112057000348.CrossRefGoogle Scholar
Crawford, D. R., Lake, B. M., Saffman, P. G. and Yuen, H. C., “Stability of weakly nonlinear deep-water waves in two and three dimensions”, J. Fluid Mech. 105 (1981) 177–191; doi:10.1017/S0022112081003169.CrossRefGoogle Scholar
Crawford, D. R., Saffman, P. G. and Yuen, H. C., “Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves”, Wave Motion 2 (1980) 1–16; doi:10.1016/0165-2125(80)90029-3.CrossRefGoogle Scholar
Curtis, C. W., Carter, J. D. and Kalisch, H., “Particle paths in nonlinear Schrödinger models in the presence of linear shear currents”, J. Fluid Mech. 855 (2018) 322–350; doi:10.1017/jfm.2018.623.CrossRefGoogle Scholar
Debiane, M. and Kharif, C., “A new limiting form for steady periodic gravity waves with surface tension on deep water”, Phys. Fluids 8 (1996) 2780–2782; doi:10.1063/1.869066.CrossRefGoogle Scholar
Dhar, A. K. and Das, K. P., “A fourth-order evolution equation for deep water surface gravity waves in the presence of wind blowing over water”, Phys. Fluids A 2 (1990) 778–183; doi:10.1063/1.857731.CrossRefGoogle Scholar
Dhar, A. K. and Kirby, J. T., “Fourth-order stability analysis for capillary-gravity waves on finite-depth currents with constant vorticity”, Phys. Fluids 35 (2023) article ID: 026601; https://doi.org/10.1063/5.0136002.CrossRefGoogle Scholar
Djordjevic, V. D. and Redekopp, L. G., “On two-dimensional packets of capillary-gravity waves”, J. Fluid Mech. 79 (1977) 703–714; doi:10.1017/S0022112077000408.CrossRefGoogle Scholar
Dysthe, K. B., “Note on a modification to the nonlinear Schrödinger equation for application to deep water waves”, Proc. Roy. Soc. London Sect. A 369(1736) (1979) 105–114; doi:10.1098/rspa.1979.0154.Google Scholar
Harrison, W. J., “The influence of viscosity and capillarity on waves of finite amplitude”, Proc. London Math. Soc. (3) s2-7(1) (1909) 107–121; doi:10.1112/plms/s2-7.1.107.CrossRefGoogle Scholar
Hogan, S. J., “Some effects of surface tension on steep water waves. Part 2”, J. Fluid Mech. 96 (1980) 417–445; doi:10.1017/S0022112080002200.CrossRefGoogle Scholar
Hogan, S. J., “The fourth-order evolution equation for deep-water gravity-capillary waves”, Proc. Roy. Soc. London Sect. A 402(1823) (1985) 359–372; doi:10.1098/rspa.1985.0122.Google Scholar
Hsu, H. C., Kharif, C., Abid, M. and Chen, Y. Y., “A nonlinear Schrödinger equation for gravity–capillary water waves on arbitrary depth with constant vorticity. Part 1”, J. Fluid Mech. 854 (2018) 146–163; doi:10.1017/jfm.2018.627.CrossRefGoogle Scholar
Hur, V. M., “Shallow water models with constant vorticity”, Eur. J. Mech. B Fluids 73 (2019) 170–179; doi:10.1016/j.euromechflu.2017.06.001.CrossRefGoogle Scholar
Janssen, P. A. E. M., “On a fourth-order envelope equation for deep-water waves”, J. Fluid Mech. 126 (1983) 1–11; doi:10.1017/S0022112083000014.CrossRefGoogle Scholar
Janssen, P. A. E. M., “Nonlinear four-wave interactions and freak waves”, J. Phys. Oceanogr. 33 (2003) 863–884; doi:10.1175/1520-0485(2003).2.0.CO;2>CrossRefGoogle Scholar
Kawahara, T., “Nonlinear self-modulation of capillary-gravity waves on liquid layer”, J. Phys. Soc. Japan 38 (1975) 265–270; doi:10.1143/JPSJ.38.265.CrossRefGoogle Scholar
Liao, B., Dong, G., Ma, Y. and Gao, J. L., “Linear-shear-current modified Schrödinger equation for gravity waves in finite water depth”, Phys. Rev. E (3) 96 (2017) article ID: 043111; doi:10.1103/PhysRevE.96.043111.CrossRefGoogle ScholarPubMed
Lighthill, M. J., “Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes”, J. Fluid Mech. 34 (1968) 113–143; doi:10.1017/S0022112068001795.CrossRefGoogle Scholar
Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics”, Proc. Roy. Soc. London Sect. A 360(1703) (1978) 471–488; doi:10.1098/rspa.1978.0080.Google Scholar
Longuet-Higgins, M. S., “The instabilities of gravity waves of finite amplitude in deep water II. Subharmonics”, Proc. Roy. Soc. London Sect. A 360(1703) (1978) 489–505; doi:10.1098/rspa.1978.0081.Google Scholar
MacIver, R. D., Simons, R. R. and Thomas, G. P., “Gravity waves interacting with a narrow jet-like current”, J. Geophys. Res. Oceans 111(C3) (2006); doi:10.1029/2005JC003030.CrossRefGoogle Scholar
Mei, C. C. and Lo, E., “The effects of a jet-like current on gravity waves in shallow water”, J. Phys. Oceanogr. 14 (1984) 471–477; doi:10.1175/1520-0485(1984)014<0471:TEOAJL>2.0.CO;2.2.0.CO;2>CrossRefGoogle Scholar
Miles, J. W., “A note on surface waves generated by shear-flow instability”, J. Fluid Mech. 447 (2001) 173–177; doi:10.1017/S0022112001005833.CrossRefGoogle Scholar
Onorato, M., Osborne, A. R., Serio, M. and Bertone, S., “Freak waves in random oceanic sea states”, Phys. Rev. Lett. 86(25) (2001) 5831–5834; doi:10.1103/PhysRevLett.86.5831.CrossRefGoogle ScholarPubMed
Onorato, M., Osborne, A. R., Serio, M., Cavaleri, L., Brandini, C. and Stansberg, C. T., “Extreme waves, modulational instability and second order theory: wave flume experiments on irregular waves”, Eur. J. Mech. B Fluids 25 (2006) 586–601; doi:10.1016/j.euromechflu.2006.01.002.CrossRefGoogle Scholar
Osborne, A. R., Onorato, M. and Serio, M., “The nonlinear dynamics of rogue waves and holes in deep-water gravity wave trains”, Phys. Lett. A 275 (2000) 386–393; doi:10.1016/S0375-9601(00)00575-2.CrossRefGoogle Scholar
Pal, T. and Dhar, A. K., “Nonlinear self-modulation of gravity-capillary waves on shear currents in finite depth”, ANZIAM J. 65 (2023) 248–272; doi:10.1017/S1446181123000196.CrossRefGoogle Scholar
Peregrine, D. H., “Water waves, nonlinear Schrödinger equations and their solutions”, ANZIAM J. 25 (1983) 16–43; doi:10.1017/S0334270000003891.Google Scholar
Saffman, P. G. and Yuen, H. C., “Bifurcation and symmetry breaking in nonlinear dispersive waves”, Phys. Rev. Lett. 44 (1980) article ID: 1097; doi:10.1103/PhysRevLett.44.1097.CrossRefGoogle Scholar
Saffman, P. G. and Yuen, H. C., “A new type of three-dimensional deep-water wave of permanent form”, J. Fluid Mech. 101 (1980) 797–808; doi:10.1017/S0022112080001930.CrossRefGoogle Scholar
Sekerzh-Zenkovich, Y. I., “On the theory of stationary capillary waves of finite amplitude on the surface of a heavy fluid”, Doklady Akad. Nauk. SSSR 109 (1956) 913–918.Google Scholar
Shrira, V. I., “Surface waves on shear currents: solution of the boundary-value problem”, J. Fluid Mech. 252 (1993) 565–584; doi:10.1017/S002211209300388X.CrossRefGoogle Scholar
Stiassnie, M., “Note on the modified nonlinear Schrödinger equation for deep water waves”, Wave Motion 6 (1984) 431–433; doi:10.1016/0165-2125(84)90043-X.CrossRefGoogle Scholar
Thomas, R., Kharif, C. and Manna, M., “A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity”, Phys. Fluids 24 (2012) article ID: 127102; doi:10.1063/1.4768530.CrossRefGoogle Scholar
Trulsen, K. and Dysthe, K. B., “A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water”, Wave Motion 24 (1996) 281–289; doi:10.1016/S0165-2125(96)00020-0.CrossRefGoogle Scholar
Whitham, G. B., “Non-linear dispersion of water waves”, J. Fluid Mech. 27 (1967) 399–412; doi:10.1017/S0022112067000424.CrossRefGoogle Scholar
Wilton, J. R., “LXXII. On ripples”, London Edinburgh Dublin Philos. Mag. J. Sci. 29 (173) (1915) 688–700; doi:10.1080/14786440508635350.CrossRefGoogle Scholar
Zakharov, V. E., “Stability of periodic waves of finite amplitude on the surface of a deep fluid”, J. Appl. Mech. Tech. Phys. 9 (1968) 190–194; doi:10.1007/BF00913182.CrossRefGoogle Scholar
Zakharov, V. E., “Collapse of Langmuir waves”, Sov. Phys. JETP 35 (1972) 908–914; https://www-thphys.physics.ox.ac.uk/people/AlexanderSchekochihin/notes/PlasmaClassics/zakharov72.pdf.Google Scholar
Zhang, X., “Short surface waves on surface shear”, J. Fluid Mech. 541 (2005) 345–370; doi:10.1017/S0022112005006063.CrossRefGoogle Scholar