Skip to main content
Top

2019 | OriginalPaper | Chapter

Gravity-Capillary and Flexural-Gravity Solitary Waves

Authors : Emilian I. Părău, Jean-Marc Vanden-Broeck

Published in: Nonlinear Water Waves

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

Solitary gravity-capillary and flexural-gravity waves in two and three dimensions of space are reviewed in this paper. Numerical methods used to compute the solitary waves are described in detail and typical solutions found over the years are presented. Similarities and differences between the solutions for the two physical problems are discussed.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literature
2.
go back to reference B. Akers, P.A. Milewski, A model equation for wavepacket solitary waves arising from capillary-gravity flows. Stud. Appl. Math. 122(3), 249–274 (2009)CrossRefMathSciNetMATH B. Akers, P.A. Milewski, A model equation for wavepacket solitary waves arising from capillary-gravity flows. Stud. Appl. Math. 122(3), 249–274 (2009)CrossRefMathSciNetMATH
3.
go back to reference T.R. Akylas, Envelope solitons with stationary crests. Phys. Fluids A 5(4), 789–791 (1993)CrossRefMATH T.R. Akylas, Envelope solitons with stationary crests. Phys. Fluids A 5(4), 789–791 (1993)CrossRefMATH
5.
go back to reference T.R. Akylas, Y. Cho, On the stability of lumps and wave collapse in water waves. Philos. Trans. R. Soc. A 366(1876), 2761–2774 (2008)CrossRefMathSciNetMATH T.R. Akylas, Y. Cho, On the stability of lumps and wave collapse in water waves. Philos. Trans. R. Soc. A 366(1876), 2761–2774 (2008)CrossRefMathSciNetMATH
7.
8.
go back to reference K.M. Berger, P.A. Milewski, The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Math. 61(3), 731–750 (2000)CrossRefMathSciNetMATH K.M. Berger, P.A. Milewski, The generation and evolution of lump solitary waves in surface-tension-dominated flows. SIAM J. Appl. Math. 61(3), 731–750 (2000)CrossRefMathSciNetMATH
9.
go back to reference B. Buffoni, M.D. Groves, S.-M. Sun, E. Wahlén, Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves. J. Differ. Equ. 254, 1006–1096 (2013)CrossRefMathSciNetMATH B. Buffoni, M.D. Groves, S.-M. Sun, E. Wahlén, Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves. J. Differ. Equ. 254, 1006–1096 (2013)CrossRefMathSciNetMATH
10.
go back to reference B. Buffoni, M.D. Groves, E. Wahlén, A variational reduction and the existence of a fully localised solitary wave for the three-dimensional water-wave problem with weak surface tension. Arch. Rat. Mech. Anal. 228, 773–820 (2018)CrossRefMathSciNetMATH B. Buffoni, M.D. Groves, E. Wahlén, A variational reduction and the existence of a fully localised solitary wave for the three-dimensional water-wave problem with weak surface tension. Arch. Rat. Mech. Anal. 228, 773–820 (2018)CrossRefMathSciNetMATH
11.
go back to reference A.R. Champneys, J.-M. Vanden-Broeck, G.J. Lord, Do true elevation gravity-capillary solitary waves exist? A numerical investigation. J. Fluid Mech. 454, 403–417 (2002)CrossRefMathSciNetMATH A.R. Champneys, J.-M. Vanden-Broeck, G.J. Lord, Do true elevation gravity-capillary solitary waves exist? A numerical investigation. J. Fluid Mech. 454, 403–417 (2002)CrossRefMathSciNetMATH
12.
go back to reference D. Clamond, D. Dutykh, A. Durán, A plethora of generalised solitary gravity-capillary water waves. J. Fluid Mech. 784, 664–680 (2015)CrossRefMathSciNetMATH D. Clamond, D. Dutykh, A. Durán, A plethora of generalised solitary gravity-capillary water waves. J. Fluid Mech. 784, 664–680 (2015)CrossRefMathSciNetMATH
16.
go back to reference F. Dias, G. Iooss, Water-waves as a spatial dynamical system, in Handbook of Mathematical Fluid Dynamics, vol. 2 (North-Holland, Amsterdam, 2003), pp. 443–499MATH F. Dias, G. Iooss, Water-waves as a spatial dynamical system, in Handbook of Mathematical Fluid Dynamics, vol. 2 (North-Holland, Amsterdam, 2003), pp. 443–499MATH
17.
18.
go back to reference F. Dias, P. Milewski, On the fully-nonlinear shallow-water generalized Serre equations. Phys. Lett. A 374(8), 1049–1053 (2010)CrossRefMATH F. Dias, P. Milewski, On the fully-nonlinear shallow-water generalized Serre equations. Phys. Lett. A 374(8), 1049–1053 (2010)CrossRefMATH
19.
go back to reference F. Dias, D. Menasce, J.-M. Vanden-Broeck, Numerical study of capillary-gravity solitary waves. Eur. J. Mech. B/Fluids 15, 17–36 (1996)MathSciNetMATH F. Dias, D. Menasce, J.-M. Vanden-Broeck, Numerical study of capillary-gravity solitary waves. Eur. J. Mech. B/Fluids 15, 17–36 (1996)MathSciNetMATH
20.
21.
go back to reference A.I. Dyachenko, E.A. Kuznetsov, M.D. Spector, V.E. Zakharov, Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 73–79 (1996)CrossRef A.I. Dyachenko, E.A. Kuznetsov, M.D. Spector, V.E. Zakharov, Analytical description of the free surface dynamics of an ideal fluid (canonical formalism and conformal mapping). Phys. Lett. A 221, 73–79 (1996)CrossRef
22.
go back to reference L.K. Forbes, Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409–428 (1986) L.K. Forbes, Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution. J. Fluid Mech. 169, 409–428 (1986)
23.
go back to reference L.K. Forbes, Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solution. J. Fluid Mech. 188, 491–508 (1988) L.K. Forbes, Surface waves of large amplitude beneath an elastic sheet. Part 2. Galerkin solution. J. Fluid Mech. 188, 491–508 (1988)
24.
25.
go back to reference T. Gao, J.-M. Vanden-Broeck, Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves. Phys. Fluids 26, 087101 (2014)CrossRefMATH T. Gao, J.-M. Vanden-Broeck, Numerical studies of two-dimensional hydroelastic periodic and generalised solitary waves. Phys. Fluids 26, 087101 (2014)CrossRefMATH
26.
go back to reference T. Gao, Z. Wang, J.-M. Vanden-Broeck. New hydroelastic solitary waves in deep water and their dynamics. J. Fluid Mech. 788, 469–491 (2016)CrossRefMathSciNetMATH T. Gao, Z. Wang, J.-M. Vanden-Broeck. New hydroelastic solitary waves in deep water and their dynamics. J. Fluid Mech. 788, 469–491 (2016)CrossRefMathSciNetMATH
27.
go back to reference T. Gao, Z. Wang, J.-M. Vanden-Broeck, On asymmetric generalized solitary gravity-capillary waves in finite depth. Proc. R. Soc. A 472, 20160454 (2016)CrossRefMathSciNetMATH T. Gao, Z. Wang, J.-M. Vanden-Broeck, On asymmetric generalized solitary gravity-capillary waves in finite depth. Proc. R. Soc. A 472, 20160454 (2016)CrossRefMathSciNetMATH
28.
go back to reference T. Gao, J.-M. Vanden-Broeck, Z. Wang, Numerical computations of two-dimensional flexural-gravity solitary waves on water of arbitrary depth. IMA J. Appl. Math. 83, 436–450 (2018)CrossRefMathSciNetMATH T. Gao, J.-M. Vanden-Broeck, Z. Wang, Numerical computations of two-dimensional flexural-gravity solitary waves on water of arbitrary depth. IMA J. Appl. Math. 83, 436–450 (2018)CrossRefMathSciNetMATH
30.
go back to reference R. Grimshaw, B. Malomed, E. Benilov, Solitary waves with damped oscillatory tails: an analysis of the fifth-order Korteweg-de Vries equation. Phys. D 77, 473–485 R. Grimshaw, B. Malomed, E. Benilov, Solitary waves with damped oscillatory tails: an analysis of the fifth-order Korteweg-de Vries equation. Phys. D 77, 473–485
31.
go back to reference M.D. Groves, S.-M. Sun, Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem. Arch. Rat. Mech. Anal. 188, 1–91 (2008)CrossRefMathSciNetMATH M.D. Groves, S.-M. Sun, Fully localised solitary-wave solutions of the three-dimensional gravity-capillary water-wave problem. Arch. Rat. Mech. Anal. 188, 1–91 (2008)CrossRefMathSciNetMATH
32.
go back to reference M.D. Groves, B. Hewer, E. Wahlén, Variational existence theory for hydroelastic solitary waves. C. R. Math. Acad. Sci. Paris 354, 1078–1086 (2016)CrossRefMathSciNetMATH M.D. Groves, B. Hewer, E. Wahlén, Variational existence theory for hydroelastic solitary waves. C. R. Math. Acad. Sci. Paris 354, 1078–1086 (2016)CrossRefMathSciNetMATH
33.
go back to reference P. Guyenne, E.I. Părău, Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713, 307–329 (2012)CrossRefMathSciNetMATH P. Guyenne, E.I. Părău, Computations of fully nonlinear hydroelastic solitary waves on deep water. J. Fluid Mech. 713, 307–329 (2012)CrossRefMathSciNetMATH
34.
go back to reference P. Guyenne, E.I. Părău, Finite-depth effects on solitary waves in a floating ice sheet. J. Fluids Struct. 49, 242–262 (2014)CrossRef P. Guyenne, E.I. Părău, Finite-depth effects on solitary waves in a floating ice sheet. J. Fluids Struct. 49, 242–262 (2014)CrossRef
35.
go back to reference P. Guyenne, E.I. Părău, Forced and unforced flexural-gravity solitary waves. Proc. IUTAM 11, 44–57 (2014)CrossRef P. Guyenne, E.I. Părău, Forced and unforced flexural-gravity solitary waves. Proc. IUTAM 11, 44–57 (2014)CrossRef
36.
go back to reference P. Guyenne, E.I. Părău, Asymptotic modeling and numerical simulation of solitary waves in a floating ice sheet, in Proceedings of 25th International Ocean Polar Engineering Conference (ISOPE 2015), Kona, Hawaii, 21–26 June 2015, pp. 467–475 P. Guyenne, E.I. Părău, Asymptotic modeling and numerical simulation of solitary waves in a floating ice sheet, in Proceedings of 25th International Ocean Polar Engineering Conference (ISOPE 2015), Kona, Hawaii, 21–26 June 2015, pp. 467–475
37.
go back to reference M. Hărăguş-Courcelle, A. Il’ichev, Three-dimensional solitary waves in the presence of additional surface effects. Eur. J. Mech. B/Fluids 17(5), 739–768 (1998)CrossRefMathSciNetMATH M. Hărăguş-Courcelle, A. Il’ichev, Three-dimensional solitary waves in the presence of additional surface effects. Eur. J. Mech. B/Fluids 17(5), 739–768 (1998)CrossRefMathSciNetMATH
38.
go back to reference J.K. Hunter, J.-M. Vanden-Broeck, Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219 (1983)CrossRefMathSciNetMATH J.K. Hunter, J.-M. Vanden-Broeck, Solitary and periodic gravity-capillary waves of finite amplitude. J. Fluid Mech. 134, 205–219 (1983)CrossRefMathSciNetMATH
39.
go back to reference A.T. Il’ichev, V.J. Tomashpolskii, Characteristic parameters of nonlinear surface envelope waves beneath an ice cover under pre-stress. Wave Motion 86, 11–20 (2019)CrossRefMathSciNet A.T. Il’ichev, V.J. Tomashpolskii, Characteristic parameters of nonlinear surface envelope waves beneath an ice cover under pre-stress. Wave Motion 86, 11–20 (2019)CrossRefMathSciNet
40.
go back to reference G. Iooss, K. Kirchgässner, Bifurcation d’ondes solitaires en présence d’une faible tension superficielle. C. R. Acad. Sci. Paris Ser. 1 311, 265–268 (1990)MATH G. Iooss, K. Kirchgässner, Bifurcation d’ondes solitaires en présence d’une faible tension superficielle. C. R. Acad. Sci. Paris Ser. 1 311, 265–268 (1990)MATH
41.
go back to reference G. Iooss, K. Kirchgässner, Water waves for small surface tension: an approach via normal form. Proc. R. Soc. Edin. A 122, 267–299 (1992)CrossRefMathSciNetMATH G. Iooss, K. Kirchgässner, Water waves for small surface tension: an approach via normal form. Proc. R. Soc. Edin. A 122, 267–299 (1992)CrossRefMathSciNetMATH
42.
go back to reference G. Iooss, P. Kirrmann, Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves. Arch. Rat. Mech. Anal. 136, 1–19 (1996)CrossRefMATH G. Iooss, P. Kirrmann, Capillary gravity waves on the free surface of an inviscid fluid of infinite depth. Existence of solitary waves. Arch. Rat. Mech. Anal. 136, 1–19 (1996)CrossRefMATH
43.
go back to reference B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15(6), 539–541 (1970)MATH B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersing media. Sov. Phys. Dokl. 15(6), 539–541 (1970)MATH
45.
go back to reference K. Kirchgässner, Nonlinear resonant surface waves and homoclinic bifurcation. Adv. Appl. Math. 26, 135–181 (1988)MATH K. Kirchgässner, Nonlinear resonant surface waves and homoclinic bifurcation. Adv. Appl. Math. 26, 135–181 (1988)MATH
46.
go back to reference A. Korobkin, E.I. Părău, J.-M. Vanden-Broeck, The mathematical challenges and modelling of the hydroelasticity. Philos. Trans. Royal Soc. A. 369, 2803–2812 (2011)CrossRefMathSciNetMATH A. Korobkin, E.I. Părău, J.-M. Vanden-Broeck, The mathematical challenges and modelling of the hydroelasticity. Philos. Trans. Royal Soc. A. 369, 2803–2812 (2011)CrossRefMathSciNetMATH
47.
go back to reference D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 36, 422–433 (1895)CrossRefMathSciNetMATH D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves. Philos. Mag. 36, 422–433 (1895)CrossRefMathSciNetMATH
49.
go back to reference M.S. Longuet-Higgins, Capillary-gravity waves of solitary type and envelope solitons on deep water. J. Fluid Mech. 252, 703–711 (1993)CrossRefMathSciNetMATH M.S. Longuet-Higgins, Capillary-gravity waves of solitary type and envelope solitons on deep water. J. Fluid Mech. 252, 703–711 (1993)CrossRefMathSciNetMATH
51.
go back to reference P.A. Milewski, J.-M. Vanden-Broeck, Z. Wang, Dynamics of steep two-dimensional gravity-capillary solitary waves. J. Fluid Mech. 664, 466–477 (2010)CrossRefMathSciNetMATH P.A. Milewski, J.-M. Vanden-Broeck, Z. Wang, Dynamics of steep two-dimensional gravity-capillary solitary waves. J. Fluid Mech. 664, 466–477 (2010)CrossRefMathSciNetMATH
52.
53.
55.
go back to reference E. Părău, F. Dias, Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech. 460, 281–305 (2002)CrossRefMathSciNetMATH E. Părău, F. Dias, Nonlinear effects in the response of a floating ice plate to a moving load. J. Fluid Mech. 460, 281–305 (2002)CrossRefMathSciNetMATH
56.
go back to reference E.I. Părău, J.-M. Vanden-Broeck, Nonlinear two- and three- dimensional free surface flows due to moving disturbances. Eur. J. Mech. B/Fluids 21, 643–656 (2002)CrossRefMathSciNetMATH E.I. Părău, J.-M. Vanden-Broeck, Nonlinear two- and three- dimensional free surface flows due to moving disturbances. Eur. J. Mech. B/Fluids 21, 643–656 (2002)CrossRefMathSciNetMATH
57.
go back to reference E.I. Părău, J.-M. Vanden-Broeck, Three-dimensional waves beneath an ice sheet due to a steadily moving pressure. Philos. Trans. R. Soc. A 369, 2973–2988 (2011)CrossRefMathSciNetMATH E.I. Părău, J.-M. Vanden-Broeck, Three-dimensional waves beneath an ice sheet due to a steadily moving pressure. Philos. Trans. R. Soc. A 369, 2973–2988 (2011)CrossRefMathSciNetMATH
58.
go back to reference E.I. Părău, J.-M. Vanden-Broeck, Three-dimensional nonlinear waves under an ice sheet and related flows, in Proceedings of 21st International Offshore Polar Engineering Conference (ISOPE-2011), Maui, 19–24 June 2011 (International Society of Offshore and Polar Engineers (ISOPE), Mountain View, 2011) E.I. Părău, J.-M. Vanden-Broeck, Three-dimensional nonlinear waves under an ice sheet and related flows, in Proceedings of 21st International Offshore Polar Engineering Conference (ISOPE-2011), Maui, 19–24 June 2011 (International Society of Offshore and Polar Engineers (ISOPE), Mountain View, 2011)
59.
go back to reference E.I. Părău, J.-M. Vanden-Broeck, M.J. Cooker, Nonlinear three-dimensional gravity-capillary solitary waves. J. Fluid Mech. 536, 99–105 (2005)CrossRefMathSciNetMATH E.I. Părău, J.-M. Vanden-Broeck, M.J. Cooker, Nonlinear three-dimensional gravity-capillary solitary waves. J. Fluid Mech. 536, 99–105 (2005)CrossRefMathSciNetMATH
60.
go back to reference E.I. Părău, J.-M. Vanden-Broeck, M.J. Cooker, Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems. Phys. Fluids. 7, 122101 (2005)CrossRefMathSciNetMATH E.I. Părău, J.-M. Vanden-Broeck, M.J. Cooker, Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems. Phys. Fluids. 7, 122101 (2005)CrossRefMathSciNetMATH
62.
go back to reference F. Smith, A. Korobkin, E. Parau, D. Feltham, V. Squire, Modelling of sea-ice phenomena. Philos. Trans. R. Soc. A 376, 20180157 (2018)CrossRef F. Smith, A. Korobkin, E. Parau, D. Feltham, V. Squire, Modelling of sea-ice phenomena. Philos. Trans. R. Soc. A 376, 20180157 (2018)CrossRef
63.
go back to reference O. Trichtchenko, E.I. Parau, J.-M. Vanden-Broeck, P. Milewski, Solitary flexural-gravity waves in three dimensions. Philos. Trans. R. Soc. A 376(2129), 20170345 (2018) O. Trichtchenko, E.I. Parau, J.-M. Vanden-Broeck, P. Milewski, Solitary flexural-gravity waves in three dimensions. Philos. Trans. R. Soc. A 376(2129), 20170345 (2018)
64.
go back to reference J.-M. Vanden-Broeck, Elevation solitary waves with surface tension. Phys. Fluids A 3, 2659–2663 (1991)CrossRefMATH J.-M. Vanden-Broeck, Elevation solitary waves with surface tension. Phys. Fluids A 3, 2659–2663 (1991)CrossRefMATH
65.
go back to reference J.-M. Vanden-Broeck, Gravity-Capillary Free-Surface Flows (Cambridge University Press, Cambridge, 2010)CrossRefMATH J.-M. Vanden-Broeck, Gravity-Capillary Free-Surface Flows (Cambridge University Press, Cambridge, 2010)CrossRefMATH
66.
go back to reference J.-M. Vanden-Broeck, F. Dias, Gravity-capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240, 549–555 (1992)CrossRefMathSciNetMATH J.-M. Vanden-Broeck, F. Dias, Gravity-capillary solitary waves in water of infinite depth and related free-surface flows. J. Fluid Mech. 240, 549–555 (1992)CrossRefMathSciNetMATH
67.
go back to reference J.-M. Vanden-Broeck, E.I. Părău, Two-dimensional generalised solitary waves and periodic waves under an ice sheet. Philos. Trans. R. Soc. A. 369, 2957–2972 (2011)CrossRefMATH J.-M. Vanden-Broeck, E.I. Părău, Two-dimensional generalised solitary waves and periodic waves under an ice sheet. Philos. Trans. R. Soc. A. 369, 2957–2972 (2011)CrossRefMATH
68.
69.
go back to reference Z. Wang, J.-M. Vanden-Broeck, Multilump symmetric and nonsymmetric gravity-capillary solitary waves in deep water. SIAM J. Appl. Math. 75, 978–998 (2015)CrossRefMathSciNetMATH Z. Wang, J.-M. Vanden-Broeck, Multilump symmetric and nonsymmetric gravity-capillary solitary waves in deep water. SIAM J. Appl. Math. 75, 978–998 (2015)CrossRefMathSciNetMATH
70.
go back to reference Z. Wang, J.-M. Vanden-Broeck, P.A. Milewski, Two-dimensional flexural-gravity waves of finite amplitude in deep water. IMA J. Appl. Math. 78, 750–761 (2013)CrossRefMathSciNetMATH Z. Wang, J.-M. Vanden-Broeck, P.A. Milewski, Two-dimensional flexural-gravity waves of finite amplitude in deep water. IMA J. Appl. Math. 78, 750–761 (2013)CrossRefMathSciNetMATH
71.
go back to reference Z. Wang, P.A. Milewski, J.-M. Vanden-Broeck, Computation of three-dimensional flexural-gravity solitary waves in arbitrary depth. Proc. IUTAM 11, 119–129 (2014)CrossRef Z. Wang, P.A. Milewski, J.-M. Vanden-Broeck, Computation of three-dimensional flexural-gravity solitary waves in arbitrary depth. Proc. IUTAM 11, 119–129 (2014)CrossRef
72.
go back to reference Z. Wang, J.-M. Vanden-Broeck, P.A. Milewski, Asymmetric gravity-capillary solitary waves on deep water. J. Fluid Mech. 759, R2 (2014)CrossRefMathSciNetMATH Z. Wang, J.-M. Vanden-Broeck, P.A. Milewski, Asymmetric gravity-capillary solitary waves on deep water. J. Fluid Mech. 759, R2 (2014)CrossRefMathSciNetMATH
73.
74.
go back to reference T.S. Yang, T.R. Akylas, On asymmetric gravity-capillary solitary waves. J. Fluid Mech. 330, 215–232 (1997)CrossRefMATH T.S. Yang, T.R. Akylas, On asymmetric gravity-capillary solitary waves. J. Fluid Mech. 330, 215–232 (1997)CrossRefMATH
Metadata
Title
Gravity-Capillary and Flexural-Gravity Solitary Waves
Authors
Emilian I. Părău
Jean-Marc Vanden-Broeck
Copyright Year
2019
DOI
https://doi.org/10.1007/978-3-030-33536-6_11

Premium Partner