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Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation

Abstract

In this contribution, we prove that small amplitude, resonant harmonic, spatially periodic traveling waves (Wilton ripples) exist in a family of weakly nonlinear PDEs which model water waves. The proof is inspired by that of Reeder and Shinbrot (Arch. Rat. Mech. Anal. 77:321–347, 1981) and complements the authors’ recent, independent result proven by a perturbative technique (Akers and Nicholls 2021). The method is based on a Banach Fixed Point Iteration and, in addition to proving that this iteration has Wilton ripples as a fixed point, we use it as a numerical method for simulating these solutions. The output of this numerical scheme and its performance are evaluated against a quasi-Newton iteration.

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References

  1. 1.

    Adams, R.A.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London. Pure Appl. Math. 65 (1975)

  2. 2.

    Akers, B., Milewski, P.: A model equation for wavepacket solitary waves arising from capillary-gravity flows. Stud. Appl. Math. 122, 249–274 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Akers, B., Nicholls, D.P.: Traveling waves with gravity and surface tension. SIAM J. Appl. Math. 70, 2373–2389 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Akers, B., Nicholls, D.P.: Spectral stability of deep two-dimensional gravity water waves: repeated eigenvalues. SIAM J. Appl. Math. 72, 689–711 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  5. 5.

    Akers, B., Nicholls, D.P.: Spectral stability of deep two-dimensional gravity capillary water waves. Stud. Appl. Math. 130, 81–107 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Akers, B., Nicholls, D.P.: The spectrum of finite depth water waves. Eur. J. Mech. B/Fluids 46, 181–189 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Akers, B., Nicholls, D.P.: Wilton ripples in weakly nonlinear dispersive models of water waves: Existence and analyticity of solution branches. Water Waves 3(1), 25–47 (2021)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Akers, B.F.: Modulational instabilities of periodic traveling waves in deep water. Physica D: Nonlinear Phenomena 300, 26–33 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Akers, B.F.: High-order perturbation of surfaces short course: Stability of travelling water waves. Lectures Theory Water Waves 426, 51 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Akers, B.F., Ambrose, D.M., Sulon, D.W.: Periodic travelling interfacial hydroelastic waves with or without mass ii: Multiple bifurcations and ripples. Eur. J. Appl. Math. 30(4), 756–790 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Akers, B.F., Gao, W.: Wilton ripples in weakly nonlinear model equations. Commun. Math. Sci. 10(3), 1015–1024 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12.

    Allgower, E.L., Georg, K.: Numerical Continuation Methods. Springer-Verlag, Berlin (1990)

    MATH  Book  Google Scholar 

  13. 13.

    Ambrose, D.M., Strauss, W.A., Wright, J.D.: Global bifurcation theory for periodic traveling interfacial gravity-capillary waves. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(4), 1081–1101 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Amundsen, D., Benney, D.: Resonances in dispersive wave systems. Stud. Appl. Math. 105(3), 277–300 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bona, J.L., Albert, J.P., Restrepo, J.M.: Solitary-wave solutions of the Benjamin equation. SIAM J. Appl. Math. 59(6), 2139–2161 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16.

    Boyd, J.P.: Chebyshev and Fourier spectral methods, 2nd edn. Dover Publications Inc., Mineola (2001)

    MATH  Google Scholar 

  17. 17.

    Bronski, J.C., Hur, V.M., Johnson, M.A.: Modulational instability in equations of KdV type. In: New Approaches to Nonlinear Waves, pp. 83–133. Springer (2016)

  18. 18.

    Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods in fluid dynamics. Springer-Verlag, New York (1988)

    MATH  Book  Google Scholar 

  19. 19.

    Christodoulides, P., Dias, F.: Resonant capillary-gravity interfacial waves. J. Fluid Mech. 265, 303–343 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Claassen, K.M., Johnson, M.A.: Numerical bifurcation and spectral stability of wavetrains in bidirectional Whitham models. Stud. Appl. Math. 141(2), 205–246 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Craig, W., Nicholls, D.P.: Traveling two and three dimensional capillary gravity water waves. SIAM J. Math. Anal. 32, 323–359 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22.

    Craig, W., Nicholls, D.P.: Traveling gravity water waves in two and three dimensions. Eur. J. Mech. B/Fluids 21, 615–641 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Craik, A.: George Gabriel Stokes on water wave theory. Ann. Rev. Fluid Mech. 37, 23–42 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Creedon, R., Deconinck, B., Trichtchenko, O.: High-frequency instabilities of a Boussinesq–Whitham system: a perturbative approach. Fluids. 6(4), 136 (2021)

  26. 26.

    Creedon, R., Deconinck, B., Trichtchenko, O.: High frequency instabilities of the Kawahara equation: A perturbative approach. preprint (2020)

  27. 27.

    Deconinck, B., Oliveras, K.: The instability of periodic surface gravity waves. J. Fluid Mech. 675, 141–167 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Deconinck, B., Trichtchenko, O.: Stability of periodic gravity waves in the presence of surface tension. Eur. J. Mech. B/Fluids 46, 97–108 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  29. 29.

    Ehrnström, M., Johnson, M.A., Maehlen, O.I., Remonato, F.: On the bifurcation diagram of the capillary-gravity Whitham equation. Water Waves 1, 275–313 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  30. 30.

    Evans, L.C.: Partial differential equations, 2nd edn. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  31. 31.

    Folland, G.B.: Introduction to partial differential equations. Preliminary informal notes of university courses and seminars in mathematics, Mathematical Notes. Princeton University Press, Princeton (1976)

  32. 32.

    Gottlieb, D., Orszag, S.A.: Numerical analysis of spectral methods: theory and applications. Society for Industrial and Applied Mathematics. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. Philadelphia, PA (1977)

  33. 33.

    Harrison, W.: The influence of viscosity and capillarity on waves of finite amplitude. Proc. Lond. Math. Soc. 2(1), 107–121 (1909)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Haupt, S.E., Boyd, J.P.: Modeling nonlinear resonance: A modification to the Stokes perturbation expansion. Wave Motion 10(1), 83–98 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Henderson, D.M., Hammack, J.L.: Experiments on ripple instabilities. Part 1. Resonant triads. J. Fluid Mech. 184, 15–41 (1987)

    Article  Google Scholar 

  36. 36.

    Hur, V.M., Johnson, M.A.: Modulational instability in the Whitham equation for water waves. Stud. Appl. Math. 134(1), 120–143 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  37. 37.

    Hur, V.M., Johnson, M.A.: Modulational instability in the Whitham equation with surface tension and vorticity. Nonlinear Anal. Theory Methods Appl. 129, 104–118 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38.

    Hur, V.M., Pandey, A.K.: Modulational instability in a full-dispersion shallow water model. Stud. Appl. Math. 142(1), 3–47 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Kamesvara, R.J.: On ripples of finite amplitude. Proc. Indian Ass. Cultiv. Sci 6, 175–193 (1920)

    Google Scholar 

  40. 40.

    Kawahara, T.: Oscillatory solitary waves in dispersive media. J. Phys. Soc. Jpn. 33(1), 260–264 (1972)

    Article  Google Scholar 

  41. 41.

    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Linear and quasilinear elliptic equations. Academic Press, New York (1968)

    MATH  Google Scholar 

  42. 42.

    Lamb, H.: Hydrodynamics, 6th edn. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  43. 43.

    Levi-Civita, T.: Détermination rigoureuse des ondes permanentes d’ampleur finie. Math. Ann. 93, 264–314 (1925)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Longuet-Higgins, M.: Some relations between Stokes’ coefficients in the theory of gravity waves. J. Inst. Math. Appl. 22, 261 (1978)

    MathSciNet  MATH  Article  Google Scholar 

  45. 45.

    McGoldrick, L.: An experiment on second-order capillary gravity resonant wave interactions. J. Fluid Mech. 40, 251–271 (1970)

    Article  Google Scholar 

  46. 46.

    McGoldrick, L.: On Wilton’s ripples: a special case of resonant interactions. J. Fluid Mech. 42(1), 193–200 (1970)

    MATH  Article  Google Scholar 

  47. 47.

    Moldabayev, D., Kalisch, H., Dutykh, D.: The Whitham equation as a model for surface water waves. Physica D: Nonlinear Phenomena 309, 99–107 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  48. 48.

    Nicholls, D., Reitich, F.: Stable, high-order computation of traveling water waves in three dimensions. Eur. J. Mech. B/Fluids 25, 406–424 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Nicholls, D.P., Reitich, F.: On analyticity of traveling water waves. Proc. R. Soc. Lond. A 461(2057), 1283–1309 (2005)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Perlin, M., Henderson, D., Hammack, J.: Experiments on ripple instabilities. Part 2 selective amplification of resonant triads. J. Fluid Mech. 219, 51–80 (1990)

    Article  Google Scholar 

  51. 51.

    Reeder, J., Shinbrot, M.: On Wilton ripples II: Rigorous results. Arch. Rat. Mech. Anal. 77, 321–347 (1981)

    MathSciNet  MATH  Article  Google Scholar 

  52. 52.

    Schwartz, L., Vanden-Broeck, J.M.: Numerical solution of the exact equations for capillary-gravity waves. J. Fluid Mech. 95, 119 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Schwartz, L.W.: Computer extension and analytic continuation of Stokes expansion for gravity waves. J. Fluid Mech. 62(3), 553–578 (1974)

    MATH  Article  Google Scholar 

  54. 54.

    Shen, J., Tang, T.: Spectral and high-order methods with applications, Mathematics Monograph Series, vol. 3. Science Press Beijing, Beijing (2006)

    Google Scholar 

  55. 55.

    Shen, J., Tang, T., Wang, L.L.: Spectral methods. Springer Series in Computational Mathematics. Algorithms, analysis and applications, vol. 41. Springer, Heidelberg (2011)

  56. 56.

    Stokes, G.G.: On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8, 441–455 (1847)

    Google Scholar 

  57. 57.

    Struik, D.: Détermination rigoureuse des ondes irrotationnelles périodiques dans un canal à profondeur finie. Math. Ann. 95, 595–634 (1926)

    MathSciNet  MATH  Article  Google Scholar 

  58. 58.

    Toland, J.F., Jones, M.: The bifurcation and secondary bifurcation of capillary-gravity waves. Proc. R. Soc. Lond. A Math. Phys. Sci. 399(1817), 391–417 (1985)

    MathSciNet  MATH  Google Scholar 

  59. 59.

    Trichtchenko, O., Deconinck, B., Kollár, R.: Stability of periodic traveling wave solutions to the Kawahara equation. SIAM J. Appl. Dyn. Syst. 17(4), 2761–2783 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  60. 60.

    Vanden-Broeck, J.M.: Wilton ripples generated by a moving pressure disturbance. J. Fluid Mech. 451, 193–201 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Wilton, J.: On ripples. Phil. Mag. 29, 173 (1915)

    MATH  Article  Google Scholar 

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Correspondence to David P. Nicholls.

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B.A. was supported during the preparation of this manuscript by the Air Force Office of Sponsored Research and the Office of Naval Research. D.P.N. gratefully acknowledges support from the National Science Foundation through Grant No. DMS–1813033.

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Akers, B., Nicholls, D.P. Wilton Ripples in Weakly Nonlinear Models of Water Waves: Existence and Computation. Water Waves (2021). https://doi.org/10.1007/s42286-021-00052-2

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Keywords

  • Wilton Ripples
  • weakly nonlinear PDEs
  • Whitham equation
  • Benjamin equation
  • Kawahara equation
  • Akers–Milewski equation