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Trimodal Steady Water Waves

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Abstract

We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves of finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave is, generically, a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave.

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References

  1. Baldi P., Toland J.F.: Bifurcation and secondary bifurcation of heavy periodic hydroelastic travelling waves. Interfaces Free Bound. 12(1), 1–22 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Buffoni B., Dancer E.N., Toland J.F.: The sub-harmonic bifurcation of Stokes waves. Arch. Ration. Mech. Anal. 152(3), 241–270 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  3. Constantin, A.: Nonlinear water waves with applications to wave-current interactions and tsunamis. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 81. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2011)

  4. Constantin A., Strauss W.A.: Exact steady periodic water waves with vorticity. Commun. Pure Appl. Math. 57(4), 481–527 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Constantin A., Varvaruca E.: Steady periodic water waves with constant vorticity: Regularity and local bifurcation. Arch. Ration. Mech. Anal. 199, 33–67 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Dubreil-Jacotin, M.L.: Sur la détermination rigoureuse des ondes permanentes périodiques d’ampleur finie. J. Math. Pures Appl. 9(13), 217–291 (1934)

  8. Ehrnström, M., Escher, J., Villari, G.: Steady water waves with multiple critical layers: interior dynamics. J. Math. Fluid Mech. 14(3), 407–419 (2012)

  9. Ehrnström, M., Escher, J., Wahlén, E.: Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456 (2011)

  10. Ehrnström, M., Villari, G.: Linear water waves with vorticity: rotational features and particle paths. J. Differ. Equ. 244(8), 1888–1909 (2008)

  11. Escher J., Matioc A.V., Matioc B.V.: On stratified steady periodic water waves with linear density distribution and stagnation points. J. Differ. Equ. 251, 2932–2949 (2011)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Gerstner F.: Theorie der Wellen samt einer daraus abgeleiteten Theorie der Deichprofile. Ann. Phys. 2, 412–445 (1809)

    Article  Google Scholar 

  13. Henry D., Matioc, A.V.: Global bifurcation of capillary-gravity stratified water waves. Proc. R. Soc. Edinburgh Sect. A 144A, 775–786 (2014)

  14. Jones M.C.W.: Small amplitude capillary-gravity waves in a channel of finite depth. Glasgow Math. J. 31(2), 141–160 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jones M.C.W., Toland J.F.: Symmetry and the bifurcation of capillary-gravity waves. Arch. Ration. Mech. Anal. 96(1), 29–53 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kielhö fer, H.: Bifurcation Theory, Applied Mathematical Sciences, vol. 156. Springer, New York, 2004

  17. Ko J., Strauss W.: Large-amplitude steady rotational water waves. Eur. J. Mech. B Fluids 27, 96–109 (2008)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Kozlov, V., Kuznetsov, N.: Steady free-surface vortical flows parallel to the horizontal bottom. Q. J. Mech. Appl. Math. 64, 371–399 (2011)

  19. Kozlov, V., Kuznetsov, N.: Bounds for steady water waves with vorticity. J. Differ. Equ. 252, 663–691 (2012)

  20. Kozlov V., Kuznetsov N.: Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents. Arch. Ration. Mech. Anal. 214, 971–1018 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Martin C.I., Matioc B.V.: Existence of Wilton ripples for water waves with constant vorticity and capillary effects. SIAM J. Appl. Math. 73(4), 1582–1595 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  22. Matioc B.V.: Global bifurcation for water waves with capillary effects and constant vorticity. Monatsh. Math. 174, 459–475 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  23. Okamoto, H., Shōji, M.: The mathematical theory of permanent progressive water-waves. Advanced Series in Nonlinear Dynamics, vol. 20. World Scientific Publishing Co. Inc., River Edge, (2001)

  24. Shatah J., Walsh S., Zeng C.: Travelling water waves with compactly supported vorticity. Nonlinearity 26, 1529–1564 (2013)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  25. Teles da Silva, A.F., Peregrine, D.H.: Steep, steady surface waves on water of finite depth with constant vorticity. J. Fluid Mech. 195, 281–302 (1988)

  26. Toland, J.F., Jones, M.C.W.: The bifurcation and secondary bifurcation of capillary-gravity waves. Proc. R. Soc. Lond. Ser. A 399, 391–417 (1985)

  27. Vanden-Broeck J.M.: Periodic waves with constant vorticity in water of infinite depth. IMA J. Appl. Math. 56(3), 207–217 (1996)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  28. Varvaruca E.: Singularities of Bernoulli free boundaries. Commun. Partial Differ. Equ. 31(10–12), 1451–1477 (2006)

  29. Wahlén, E.: Steady water waves with a critical layer. J. Differ. Equ. 246(6), 2468–2483 (2009)

  30. Walsh S.: Stratified steady periodic water waves. SIAM J. Math. Anal. 41(3), 1054–1105 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491, Trondheim, Norway

    Mats Ehrnström

  2. Centre for Mathematical Sciences, Lund University, PO Box 118, 221 00, Lund, Sweden

    Erik Wahlén

Authors
  1. Mats Ehrnström
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  2. Erik Wahlén
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Corresponding author

Correspondence to Mats Ehrnström.

Additional information

Communicated by P. Rabinowitz

ME was supported by Grant No. 231668 from the Norwegian Research Council; EW by Grant No. 621-2012-3753 from the Swedish Research Council.

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Ehrnström, M., Wahlén, E. Trimodal Steady Water Waves. Arch Rational Mech Anal 216, 449–471 (2015). https://doi.org/10.1007/s00205-014-0812-3

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  • DOI: https://doi.org/10.1007/s00205-014-0812-3

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