Papers
Topics
Authors
Recent
Search
2000 character limit reached

Nonlinear spatial evolution of degenerate quartets of water waves

Published 11 Mar 2024 in physics.flu-dyn, math.DS, and nlin.PS | (2403.06558v1)

Abstract: In this manuscript we investigate the Benjamin-Feir (or modulation) instability for the spatial evolution of water waves from the perspective of the discrete, spatial Zakharov equation, which captures cubically nonlinear and resonant wave interactions in deep water without restrictions on spectral bandwidth. Spatial evolution, with measurements at discrete locations, is pertinent for laboratory hydrodynamic experiments, such as in wave flumes, which rely on time-series measurements at a series of fixed gauges installed along the facility. This setting is likewise appropriate for experiments in electromagnetic and plasma waves. Through a reformulation of the problem for a degenerate quartet, we bring to bear techniques of phase-plane analysis which elucidate the full dynamics without recourse to linear stability analysis. In particular we find hitherto unexplored breather solutions and discuss the optimal transfer of energy from carrier to sidebands. Finally, we discuss the observability of such discrete solutions in light of numerical simulations.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (45)
  1. On homoclinic structure and numerically induced chaos for the nonlinear Schrodinger equation. SIAM J. Appl. Math., 50(2):339–351, 1990.
  2. Generation of periodic trains of picosecond pulses in an optical fiber: exact solutions. Sov. Phys. JETP, 62(5):894–899, 1985.
  3. First-order exact solutions of the nonlinear Schrödinger equation. Teoret. Mat. Fiz., 72(2):183–196, 1987.
  4. D. Andrade and R. Stuhlmeier. Instability of waves in deep water—a discrete hamiltonian approach. European Journal of Mechanics-B/Fluids, 101:320–336, 2023.
  5. D. Andrade and R. Stuhlmeier. The nonlinear Benjamin-Feir instability - Hamiltonian dynamics, discrete breathers, and steady solutions. Journal of Fluid Mechanics, 958:A17, 2023. doi:10.1017/jfm.2023.96.
  6. Optimal frequency conversion in the nonlinear stage of modulation instability. Opt. Express, 23(24):30861, 2015.
  7. The disintegration of wave trains on deep water Part 1. Theory. J. Fluid Mech., 27(03):417–430, 1967.
  8. D. Benney and A. Newell. The propagation of nonlinear wave envelopes. Journal of mathematics and Physics, 46(1-4):133–139, 1967.
  9. Filamentary structure of light beams in nonlinear liquids. Soviet Journal of Experimental and Theoretical Physics Letters, 3:307, 1966.
  10. N. Bogoliubov. On the theory of superfluidity. J. Phys, 11(1):23, 1947.
  11. F. Bretherton. Low frequency oscillations trapped near the equator. Tellus, 16(2):181–185, 1964.
  12. M. D. Bustamante and E. Kartashova. Effect of the dynamical phases on the nonlinear amplitudes’ evolution. Europhys. Lett., 85(3), 2009.
  13. G. Cappellini and S. Trillo. Third-order three-wave mixing in single-mode fibers: exact solutions and spatial instability effects. J. Opt. Soc. Am. B, 8(4):824, 1991.
  14. Experimental study of spatiotemporally localized surface gravity water waves. Phys. Rev. E, 86(1):016311, jul 2012.
  15. A. Chabchoub and R. H. Grimshaw. The hydrodynamic nonlinear Schrödinger equation: Space and time. Fluids, 1(3):1–10, 2016.
  16. Rogue wave observation in a water wave tank. Physical Review Letters, 106(20):204502, 2011.
  17. Anatomy of the Akhmediev breather: Cascading instability, first formation time, and Fermi-Pasta-Ulam recurrence. Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys., 92(6):1–9, 2015.
  18. A. D. D. Craik. Wave Interactions and Fluid Flows. Cambridge University Press, jan 1986.
  19. Stability of weakly nonlinear deep-water waves in two and three dimensions. J. Fluid Mech., 105:177–191, 1981.
  20. On the formation of freak waves on the surface of deep water. JETP letters, 88:307–311, 2008.
  21. K. B. Dysthe. Note on a modification to the nonlinear Schrodinger equation for application to deep water waves. Proc. R. Soc. A Math. Phys. Eng. Sci., 369:105–114, aug 1979.
  22. F. Fedele and D. Dutykh. Special solutions to a compact equation for deep-water gravity waves. Journal of Fluid Mechanics, 712:646–660, 2012.
  23. Spatial deterministic wave forecasting for nonlinear sea-states. Phys. Fluids, 33(10), 2021.
  24. Interactions of coherent structures on the surface of deep water. Fluids, 4(2):1–21, 2019.
  25. The Peregrine soliton in nonlinear fibre optics. Nat. Phys., 6(10):790–795, aug 2010.
  26. V. P. Krasitskii. On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves. J. Fluid Mech., 272:1–20, 1994.
  27. E. A. Kuznetsov. Solitons in a parametrically unstable plasma. 236:575–577, 1977.
  28. On the steady-state nearly resonant waves. J. Fluid Mech., 794:175–199, 2016.
  29. Phase velocity effects in tertiary wave interactions. J. Fluid Mech., 12(3):333–336, 1962.
  30. Y.-C. Ma. The perturbed plane-wave solutions of the cubic schrödinger equation. Studies in Applied Mathematics, 60(1):43–58, 1979.
  31. Hydrodynamic and optical waves: A common approach for unidimensional propagation. Rogue and Shock Waves in Nonlinear Dispersive Media, pages 1–22, 2016.
  32. Rogue and Shock Waves in Nonlinear Dispersive Media. Springer, 2016.
  33. D. H. Peregrine. Water waves, nonlinear Schrödinger equations and their solutions. J. Aust. Math. Soc. Ser. B. Appl. Math., 25(1):16–43, 1983.
  34. O. M. Phillips. On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions. J. Fluid Mech., 9(2):193–217, oct 1960.
  35. L. Shemer and A. Chernyshova. Spatial evolution of an initially narrow-banded wave train. J. Ocean Eng. Mar. Energy, 3(4):333–351, 2017.
  36. Evolution of a nonlinear wave field along a tank: experiments and numerical simulations based on the spatial Zakharov equation. J. Fluid Mech., 427:107–129, 2001.
  37. An experimental and numerical study of the spatial evolution of unidirectional nonlinear water-wave groups. Phys. Fluids, 14(10):3380–3390, oct 2002.
  38. On the highest non-breaking wave in a group: Fully nonlinear water wave breathers versus weakly nonlinear theory. J. Fluid Mech., 735:203–248, 2013.
  39. R. Stuhlmeier. An introduction to the Zakharov equation for modelling deep water waves. In D. Henry, editor, Nonlinear Dispersive Waves. Springer, to appear.
  40. S. Trillo and S. Wabnitz. Dynamics of the nonlinear modulational instability in optical fibers. Optics letters, 16(13):986–988, 1991.
  41. K. Trulsen and K. B. Dysthe. A modified nonlinear schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion, 24(3):281–289, 1996.
  42. Observation of modulation instability and rogue breathers on stationary periodic waves. Physical Review Research, 2(3):033528, 2020.
  43. Nonlinear Dynamics of Deep-Water Gravity Waves. In Adv. Appl. Mech., pages 68–229. Academic Press, 1982.
  44. V. Zakharov. Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys., 9(2):190–194, 1968.
  45. Kolmogorov Spectra of Turbulence I. Springer-Verlag Berlin Heidelberg, 1992.
Citations (2)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Open Problems

We haven't generated a list of open problems mentioned in this paper yet.

Generate Now

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 1 tweet with 1 like about this paper.

Sign Up for Free