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Six wave interaction equations in finite-depth gravity waves with surface tension

Published online by Cambridge University Press:  14 April 2023

Mark J. Ablowitz ,
Xu-Dan Luo Open the ORCID record for Xu-Dan Luo [Opens in a new window]  and
Ziad H. Musslimani Open the ORCID record for Ziad H. Musslimani [Opens in a new window]
Mark J. Ablowitz
Affiliation:
Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, CO 80309-0526, USA
Xu-Dan Luo
Affiliation:
Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China
Ziad H. Musslimani*
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA
*
†Email address for correspondence: musslimani@math.fsu.edu
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Abstract

Three wave resonant triad interactions in two space and one time dimensions form a well-known system of first-order quadratically nonlinear evolution equations that arise in many areas of physics. In deep water waves, they were first derived by Simmons in 1969 and later shown to be exactly solvable by Ablowitz & Haberman in 1975. Specifically, integrability was established by introducing a system of six wave interactions whose symmetry reduction leads to the well-known three wave equations. Here, it is shown that the six wave interaction and classical three wave equations satisfying triad resonance conditions in finite-depth gravity waves can be derived from the non-local integro-differential formulation of the free surface gravity wave equation with surface tension. These quadratically nonlinear six wave interaction equations and their reductions to the classical and non-local complex as well as real reverse space–time three wave interaction equations are integrable. Limits to infinite and shallow water depth are also discussed.

JFM classification

Mathematical Foundations: Hamiltonian theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 961 , 25 April 2023 , A3
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Ablowitz, M.J. 2011 Nonlinear Dispersive Waves: Asymptotic Analysis And Solitons. Cambridge University Press.CrossRefGoogle Scholar
Ablowitz, M.J. & Clarkson, P.A. 1991 Solitons, Nonlinear Evolution Equations and Inverse Scattering, vol. 149. Cambridge University Press.10.1017/CBO9780511623998CrossRefGoogle Scholar
Ablowitz, M.J., Fokas, A.S. & Musslimani, Z.H. 2006 On a new non-local formulation of water waves. J. Fluid Mech. 562, 313–343.CrossRefGoogle Scholar
Ablowitz, M.J. & Haberman, R. 1975 a Nonlinear evolution equations–two and three dimensions. Phys. Rev. Lett. 35 (18), 1185.10.1103/PhysRevLett.35.1185CrossRefGoogle Scholar
Ablowitz, M.J. & Haberman, R. 1975 b Resonantly coupled nonlinear evolution equations. J. Math. Phys. 16 (11), 2301–2305.CrossRefGoogle Scholar
Ablowitz, M.J. & Haut Terry, S. 2009 Coupled nonlinear Schrödinger equations from interfacial fluids with a free surface. Theor. Math. Phys. 159, 689–697.CrossRefGoogle Scholar
Ablowitz, M.J., Luo, X.-D. & Musslimani, Z.H. 2023 Three-wave interaction equations: classical and nonlocal. SIAM J. Math. Anal. (in press).Google Scholar
Ablowitz, M.J. & Musslimani, Z.H. 2013 Integrable nonlocal nonlinear Schrödinger equation. Phys. Rev. Lett. 110 (6), 064105.10.1103/PhysRevLett.110.064105CrossRefGoogle ScholarPubMed
Ablowitz, M.J. & Musslimani, Z.H. 2016 Inverse scattering transform for the integrable nonlocal nonlinear Schrödinger equation. Nonlinearity 29 (3), 915.CrossRefGoogle Scholar
Ablowitz, M.J. & Musslimani, Z.H. 2017 Integrable nonlocal nonlinear equations. Stud. Appl. Maths 139, 7–59.CrossRefGoogle Scholar
Ablowitz, M.J. & Musslimani, Z.H. 2019 Integrable nonlocal asymptotic reductions of physically significant nonlinear equations. J. Phys. A 52 (15), 15LT02.CrossRefGoogle Scholar
Ablowitz, M.J. & Musslimani, Z.H. 2021 Integrable space–time shifted nonlocal nonlinear equations. Phys. Lett. A 409, 127516.10.1016/j.physleta.2021.127516CrossRefGoogle Scholar
Ablowitz, M.J. & Segur, H. 1979 On the evolution of packets of water waves. J. Fluid Mech. 92 (4), 691–715.CrossRefGoogle Scholar
Ablowitz, M.J. & Segur, H. 1981 Solitons and the Inverse Scattering Transform. SIAM.10.1137/1.9781611970883CrossRefGoogle Scholar
Benjamin, T.B. & Feir, J.E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27 (3), 417–430.CrossRefGoogle Scholar
Benney, D.J. 1962 Non-linear gravity wave interactions. J. Fluid Mech. 14 (4), 577–584.10.1017/S0022112062001469CrossRefGoogle Scholar
Benney, D.J. & Newell, A.C. 1967 The propagation of nonlinear envelopes. J. Math. Phys. 46, 133–139.CrossRefGoogle Scholar
Bretherton, F.P. 1964 Resonant interactions between waves, the case of discrete oscillations. J. Fluid Mech. 20 (3), 457–479.CrossRefGoogle Scholar
Buryak, A.V., Di Trapani, P., Skryabin, D.V. & Trillo, S. 2002 Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys. Rep. 370 (2), 63–235.10.1016/S0370-1573(02)00196-5CrossRefGoogle Scholar
Case, K.M. & Chiu, S.C. 1977 Three-wave resonant interactions of gravity-capillary waves. Phys. Fluids 20 (5), 742–745.CrossRefGoogle Scholar
Craik, A.D.D. 1988 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
Dyachenko, A.I., Dyachenko, S.A., Lushnikov, P.M. & Zakharov, V.E. 2021 Short branch cut approximation in two-dimensional hydrodynamics with free surface. Proc. R. Soc. A 477 (2249), 20200811.10.1098/rspa.2020.0811CrossRefGoogle Scholar
Dyachenko, A.I., Korotkevich, A.O. & Zakharov, V.E. 2003 Decay of the monochromatic capillary wave. J. Expl Theor. Phys. Lett. 77 (9), 477–481.10.1134/1.1591973CrossRefGoogle Scholar
Dyachenko, S.A., Lushnikov, P.M. & Korotkevich, A.O. 2016 Branch cuts of stokes wave on deep water. Part I: numerical solution and padé approximation. Stud. Appl. Maths 137 (4), 419–472.10.1111/sapm.12128CrossRefGoogle Scholar
Dyachenko, A.I., Zakharov, V.E. & Kuznetsov, E.A. 1996 Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys. Rep. 22 (10), 829–840.Google Scholar
Falcon, E., Laroche, C. & Fauve, S. 2002 Observation of depression solitary surface waves on a thin fluid layer. Phys. Rev. Lett. 89 (20), 204501.CrossRefGoogle ScholarPubMed
Fokas, A.S. 2006 Integrable nonlinear evolution partial differential equations in 4+ 2 and 3+ 1 dimensions. Phys. Rev. Lett. 96 (19), 190201.CrossRefGoogle ScholarPubMed
Gardner, C.S., Greene, J.M., Kruskal, M.D. & Miura, R.M. 1967 Method for solving the Korteweg–Devries equation. Phys. Rev. Lett. 19 (19), 1095.CrossRefGoogle Scholar
Gibbons, G.W., Page, D.N. & Pope, C.N. 1990 Einstein metrics on S3, R3 and R4 bundles. Commun. Math. Phys. 127 (3), 529–553.10.1007/BF02104500CrossRefGoogle Scholar
Hammack, J.L. & Henderson, D.M. 1993 Resonant interactions among surface water waves. Annu. Rev. Fluid Mech. 25 (1), 55–97.CrossRefGoogle Scholar
Hasselmann, K. 1967 Nonlinear interactions treated by the methods of theoretical physics (with application to the generation of waves by wind). Proc. R. Soc. Lond. A 299 (1456), 77–103.Google Scholar
Henderson, D.M. & Hammack, J.L. 1987 Experiments on ripple instabilities. Part 1. Resonant triads. J. Fluid Mech. 184, 15–41.CrossRefGoogle Scholar
Kadomtsev, B.B. & Petviashvili, V.I. 1970 On the stability of solitary waves in weakly dispersing media. In Doklady Akademii Nauk, vol. 192, pp. 753–756. Russian Academy of Sciences.Google Scholar
Kaup, D.J. 1976 The three-wave interaction? A nondispersive phenomenon. Stud. Appl. Maths 55 (1), 9–44.CrossRefGoogle Scholar
Kaup, D.J. 1981 The solution of the general initial value problem for the full three dimensional three-wave resonant interaction. Physica D 3 (1–2), 374–395.CrossRefGoogle Scholar
Korotkevich, A.O., Dyachenko, A.I. & Zakharov, V.E. 2016 Numerical simulation of surface waves instability on a homogeneous grid. Physica D 321, 51–66.CrossRefGoogle Scholar
Lushnikov, P.M. 2016 Structure and location of branch point singularities for stokes waves on deep water. J. Fluid Mech. 800, 557–594.CrossRefGoogle Scholar
Mason, L.J. & Woodhouse, N.M.J. 1996 Integrability, Self-Duality, and Twistor Theory. Oxford University Press.Google Scholar
McGoldrick, L.F. 1965 Resonant interactions among capillary-gravity waves. J. Fluid Mech. 21 (2), 305–331.CrossRefGoogle Scholar
McGoldrick, L.F. 1970 An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40 (2), 251–271.CrossRefGoogle Scholar
Novikov, S., Manakov, S.V., Pitaevskii, L.P. & Zakharov, V.E. 1984 Theory of Solitons: The Inverse Scattering Method. Springer Science & Business Media.Google Scholar
Perlin, M., Henderson, D. & Hammack, J. 1990 Experiments on ripple instabilities. Part 2. Selective amplification of resonant triads. J. Fluid Mech. 219, 51–80.CrossRefGoogle Scholar
Phillips, O.M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions. J. Fluid Mech. 9 (2), 193–217.CrossRefGoogle Scholar
Phillips, O.M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.Google Scholar
Simmons, W.F. 1969 A variational method for weak resonant wave interactions. Proc. R. Soc. Lond. A 309 (1499), 551–577.Google Scholar
Stokes, G.G. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–473.Google Scholar
Ward, R.S. 1977 On self-dual gauge fields. Phys. Lett. A 61 (2), 81–82.CrossRefGoogle Scholar
Whitham, G.B. 1974 Linear and Nonlinear Waves. John Wiley & Sons.Google Scholar
Yang, B. & Yang, J. 2021 General rogue waves in the three-wave resonant interaction systems. IMA J. Appl. Maths 86 (2), 378–425.10.1093/imamat/hxab005CrossRefGoogle Scholar
Yang, B. & Yang, J. 2022 Rogue waves in (2+ 1)-dimensional three-wave resonant interactions. Physica D 432, 133160.CrossRefGoogle Scholar
Zakharov, V.E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9 (2), 190–194.CrossRefGoogle Scholar
Zakharov, V.E. 1998 Weakly nonlinear waves on the surface of an ideal finite depth fluid. Am. Math. Soc. Transl., 167–197.Google Scholar
Zakharov, V.E. & Manakov, S.V. 1975 The theory of resonance interaction of wave packets in nonlinear media. Zh. Eksp. Teor. Fiz. 69, 1654–1673.Google Scholar
Zakharov, V.E. & Shabat, A. 1972 Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP 34, 62–69.Google Scholar