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A plethora of generalised solitary gravity–capillary water waves

Published online by Cambridge University Press:  06 November 2015

Didier Clamond ,
Denys Dutykh  and
Angel Durán
Didier Clamond*
Affiliation:
Université de Nice – Sophia Antipolis, Laboratoire J. A. Dieudonné, Parc Valrose, 06108 Nice CEDEX 2, France
Denys Dutykh
Affiliation:
Université Savoie Mont Blanc, LAMA, UMR 5127 CNRS, Campus Scientifique, 73376 Le Bourget-du-Lac CEDEX, France
Angel Durán
Affiliation:
Departamento de Matemática Aplicada, E.T.S.I. Telecomunicación, Campus Miguel Delibes, Universidad de Valladolid, Paseo de Belen 15, 47011 Valladolid, Spain
*
†Email address for correspondence: [email protected]
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Abstract

The present study describes, first, an efficient algorithm for computing solutions in terms of capillary–gravity solitary waves of the irrotational Euler equations with a free surface and, second, provides numerical evidences of the existence of an infinite number of generalised solitary waves (solitary waves with undamped oscillatory wings). Using conformal mapping, the unknown fluid domain, which is to be determined, is mapped into a uniform strip of the complex plane. In the transformed domain, a Babenko-like equation is then derived and solved numerically.

JFM classification

Waves/Free-surface Flows: Capillary waves Waves/Free-surface Flows: Solitary waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 784 , 10 December 2015 , pp. 664 - 680
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Copyright
© 2015 Cambridge University Press 

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References

Babenko, K. I. 1987 Some remarks on the theory of surface waves of finite amplitude. Sov. Math. Dokl. 35, 599–603.Google Scholar
Beale, J. T. 1991 Exact solitary water waves with capillary ripples at infinity. Commun. Pure Appl. Maths 44 (2), 211–257.Google Scholar
Benilov, E.S., Grimshaw, R. & Kuznetsova, E. P. 1993 The generation of radiating waves in a singularly-perturbed Korteweg–de Vries equation. Physica D 69 (3–4), 270–278.Google Scholar
Boyd, J. P. 2000 Chebyshev and Fourier Spectral Methods, 2nd edn. Dover.Google Scholar
Boyd, J. P. 2007 Why Newton’s method is hard for travelling waves: small denominators, KAM theory, Arnold’s linear Fourier problem, non-uniqueness, constraints and erratic failure. Maths Comput. Simul. 74 (2–3), 72–81.Google Scholar
Bridges, T. J. & Donaldson, N. M. 2005 Degenerate periodic orbits and homoclinic torus bifurcation. Phys. Rev. Lett. 95 (10), 104301.CrossRefGoogle ScholarPubMed
Buffoni, B., Champneys, A. R. & Toland, J. F. 1996a Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system. J. Dynam. Differ. Equ. 8 (2), 221–279.CrossRefGoogle Scholar
Buffoni, B., Dancer, E. N. & Toland, J. F. 2000 The regularity and local bifurcation of steady periodic water waves. Arch. Rat. Mech. Anal. 152, 207–240.Google Scholar
Buffoni, B., Groves, M. D. & Toland, J. F. 1996b A plethora of solitary gravity–capillary water waves with nearly critical Bond and Froude numbers. Phil. Trans. R. Soc. Lond. A 354 (1707), 575–607.Google Scholar
Calvo, D. C. & Akylas, T. R. 1997 On the formation of bound states by interacting nonlocal solitary waves. Physica D 101 (3–4), 345–362.Google Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods Fundamentals in Single Domains. Springer.Google Scholar
Champneys, A. R. & Groves, M. D. 1997 A global investigation of solitary-wave solutions to a two-parameter model for water waves. J. Fluid Mech. 342, 199–229.Google Scholar
Champneys, A. R., Vanden-Broeck, J.-M. & Lord, G. J. 2002 Do true elevation gravity–capillary solitary waves exist? A numerical investigation. J. Fluid Mech. 454, 403–417.Google Scholar
Chardard, F., Dias, F. & Bridges, T. J. 2009 On the Maslov index of multi-pulse homoclinic orbits. Proc. R. Soc. Lond. A 465 (2109), 2897–2910.Google Scholar
Choi, W. & Camassa, R. 1999 Exact evolution equations for surface waves. J. Engng Mech. 125 (7), 756–760.Google Scholar
Clamond, D. 1999 Steady finite-amplitude waves on a horizontal seabed of arbitrary depth. J. Fluid Mech. 398, 45–60.Google Scholar
Clamond, D. 2003 Cnoidal-type surface waves in deep water. J. Fluid Mech. 489, 101–120.Google Scholar
Clamond, D. & Barthélémy, E. 1995 Experimental determination of the phase shift in the Stokes wave–solitary wave interaction. C. R. Acad. Sci. Paris II 320 (6), 277–280.Google Scholar
Clamond, D. & Dutykh, D. 2013 Fast accurate computation of the fully nonlinear solitary surface gravity waves. Comput. Fluids 84, 35–38.Google Scholar
Dias, F. & Kharif, C. 1999 Nonlinear gravity and capillary–gravity waves. Ann. Rev. Fluid Mech. 31, 301–346.Google Scholar
Dias, F., Menasce, D. & Vanden-Broeck, J.-M. 1996 Numerical study of capillary–gravity solitary waves. Eur. J. Mech. (B/Fluids) 15, 17–36.Google Scholar
Dyachenko, A. I., Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1996a Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221 (1–2), 73–79.CrossRefGoogle Scholar
Dyachenko, A. I., Zakharov, V. E. & Kuznetsov, E. A. 1996b Nonlinear dynamics of the free surface of an ideal fluid. Plasma Phys. Rep. 22 (10), 829–840.Google Scholar
Grimshaw, R. & Joshi, N. 1995 Weakly nonlocal solitary waves in a singularly perturbed Korteweg–De Vries equation. SIAM J. Appl. Math. 55 (1), 124–135.Google Scholar
Hunter, J. K. & Vanden-Broeck, J.-M. 1983 Solitary and periodic gravity–capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219.Google Scholar
Jézéquel, T., Bernard, P. & Lombardi, E.2014 Homoclinic orbits with many loops near a $0^{2}\text{i}{\it\omega}$ resonant fixed point of Hamiltonian systems. Discrete Contin. Dyn. Syst. (in press); arXiv:1401.1509.Google Scholar
Levenberg, K. 1944 A method for the solution of certain problems in least squares. Q. Appl. Maths 2, 164–168.Google Scholar
Li, Y. A., Hyman, J. M. & Choi, W. 2004 A numerical study of the exact evolution equations for surface waves in water of finite depth. Stud. Appl. Maths 113, 303–324.CrossRefGoogle Scholar
Lombardi, E. 2000 Oscillatory Integrals and Phenomena Beyond all Algebraic Orders with Applications to Homoclinic Orbits in Reversible Systems. Springer.Google Scholar
Longuet-Higgins, M. S. & Fox, J. H. 1996 Asymptotic theory for the almost-highest solitary wave. J. Fluid Mech. 317, 1–19.Google Scholar
Lourakis, M. I. A.2004 levmar: Levenberg–Marquardt nonlinear least squares algorithms in C/C++.Google Scholar
Lourakis, M. L. A. & Argyros, A. A. 2005 Is Levenberg–Marquardt the most efficient optimization algorithm for implementing bundle adjustment? In Tenth IEEE International Conference on Computer Vision (ICCV’05), vol. 1, pp. 1526–1531. IEEE.Google Scholar
Marquardt, D. W. 1963 An algorithm for least-squares estimation of nonlinear parameters. J. Soc. Ind. Appl. Maths 11 (2), 431–441.Google Scholar
McLean, J. W. 1982 Instabilities of finite-amplitude water waves. J. Fluid Mech. 114, 315–330.Google Scholar
Milewski, P., Vanden-Broeck, J.-M. & Wang, Z. 2010 Dynamics of steep two-dimensional gravity–capillary solitary waves. J. Fluid Mech. 664, 466–477.Google Scholar
Moré, J. J. 1978 The Levenberg–Marquardt algorithm: implementation and theory. In Proceedings of the Biennial Conference Held at Dundee, 28 June–1 July 1977 (ed. Watson, G. A.), pp. 105–116. Springer.Google Scholar
Nielsen, H.1999 Damping parameter in Marquardt’s method. Tech. Rep. Technical University of Denmark.Google Scholar
Nocedal, J. & Wright, S. J. 2006 Numerical Optimization, 2nd edn. Springer.Google Scholar
Okamoto, H. & Shoji, M. 2001 The Mathematical Theory of Permanent Progressive Water Waves. World Scientific.Google Scholar
Ovsyannikov, L. V. 1974 To the shallow water theory foundation. Arch. Mech. 26, 407–422.Google Scholar
Pelinovsky, D. E. & Stepanyants, Y. A. 2004 Convergence of Petviashvili’s iteration method for numerical approximation of stationary solutions of nonlinear wave equations. SIAM J. Numer. Anal. 42, 1110–1127.Google Scholar
Sun, S. M. 1991 Existence of a generalized solitary wave solution for water with positive Bond number less than 1/3. J. Math. Anal. Appl. 156 (2), 471–504.Google Scholar
Sun, S. M. & Shen, M. C. 1993 Exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave. J. Math. Anal. Appl. 172 (2), 533–566.Google Scholar
Vanden-Broeck, J.-M. 2010 Gravity–Capillary Free-Surface Flows. Cambridge University Press.Google Scholar
Yang, T.-S. & Akylas, T. R. 1996 Weakly nonlocal gravity–capillary solitary waves. Phys. Fluids 8, 1506–1514.Google Scholar