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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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