- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Higher-dimensional extended shallow water equations and resonant soliton radiation

Theodoros P. Horikis1, Dimitrios J. Frantzeskakis2, Timothy R. Marchant3,4, and Noel F. Smyth5,3

  • 1Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
  • 2Department of Physics, National and Kapodistrian University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece
  • 3School of Mathematics and Applied Statistics, University of Wollongong, Northfields Avenue, Wollongong, 2522 New South Wales, Australia
  • 4Australian Mathematical Sciences Institute, University of Melbourne, Melbourne, 3052 Victoria, Australia
  • 5School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, Scotland, United Kingdom

PDF

Phys. Rev. Fluids 6, 104401 – Published 25 October, 2021

DOI: https://doi.org/10.1103/PhysRevFluids.6.104401

Abstract

The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg–de Vries and Kadomtsev-Petviashvili equations—which include higher order nonlinear, dispersive, and nonlocal terms—are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions for the resonant radiation amplitude and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude of the undular bore.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (62)

  1. G. G. Stokes, Mathematical and Physical Papers (Cambridge University Press, Cambridge, 2009), Vols. 1–5.
  2. A. D. D. Craik, George Gabriel Stokes on water wave theory, Annu. Rev. Fluid Mech. 37, 23 (2005).
  3. Sir H. Lamb, Hydrodynamics (Dover, New York, 1945).
  4. G. B. Whitham, Linear and Nonlinear Waves (John Wiley and Sons, New York, 1974).
  5. M. J. Ablowitz, Nonlinear Dispersive Waves. Asymptotic Analysis and Solitons (Cambridge University Press, Cambridge, 2011).
  6. C. S. Gardner, J. M. Greene, M. D. Kruskal, and R. M. Miura, Method for Solving the Korteweg–de Vries Equation, Phys. Rev. Lett. 19, 1095 (1967).
  7. R. S. Johnson, Water waves and Korteweg–de Vries equations, J. Fluid Mech. 97, 701 (1980).
  8. P. G. Baines, Topographic Effects in Stratified Flows, Cambridge Monographs on Mechanics (Cambridge, University Press, Cambridge, 1995).
  9. Y. A. Berezin and V. I. Karpman, Theory of non-stationary finite amplitude waves in a low density plasma, Sov. Phys. JETP 19, 1265 (1964).
  10. A. Jeffrey, The role of the Korteweg–de Vries equation in plasma physics, Q. J. R. Astron. Soc. 14, 183 (1973).
  11. Yu. S. Kivshar, Dark-soliton dynamics and shock waves induced by the stimulated Raman effect in optical fibers, Phys. Rev. A 42, 1757 (1990).
  12. D. J. Frantzeskakis, Small-amplitude solitary structures for an extended nonlinear Schrödinger equation, J. Phys. A: Math. Gen. 29, 3631 (1996).
  13. G. Huang, M. G. Velarde, and V. A. Makarov, Dark solitons and their head-on collisions in Bose-Einstein condensates, Phys. Rev. A 64, 013617 (2001).
  14. T. P. Horikis, Small-amplitude defocusing nematicons, J. Phys. A 48, 02FT01 (2015).
  15. R. Carretero-González, J. Cuevas-Maraver, D. J. Frantzeskakis, T. P. Horikis, P. G. Kevrekidis, and A. S. Rodrigues, A Korteweg–de Vries description of dark solitons in polariton superfluids, Phys. Lett. A 381, 3805 (2017).
  16. T. R. Marchant and N. F. Smyth, The extended Korteweg–de Vries equation and the resonant flow of a fluid over topography, J. Fluid Mech. 221, 263 (1990).
  17. T. R. Marchant and N. F. Smyth, An undular bore solution for the higher-order Korteweg–de Vries equation, J. Phys. A: Math. Gen. 39, L563 (2006).
  18. C. G. Hooper, P. D. Ruiz, J. M. Huntley, and K. R. Khusnutdinova, Undular bores generated by fracture, Phys. Rev. E 104, 044207 (2021).
  19. M. D. Albalwi, T. R. Marchant, and N. F. Smyth, Higher-order modulation theory for resonant flow over topography, Phys. Fluids 29, 077101 (2017).
  20. G. N. Koutsokostas, T. P. Horikis, P. G. Kevrekidis, and D. J. Frantzeskakis, Universal reductions and solitary waves of weakly nonlocal defocusing nonlinear Schrödinger equations, J. Phys. A: Math. Theor. 54, 085702 (2021).
  21. R. Grimshaw, in Environmental Stratified Flows, edited by R. Grimshaw (Kluwer, Dordrect, 2002), pp. 1–27.
  22. J. R. Apel, L. A. Ostrovsky, Y. A. Stepanyants, and J. F. Lynch, Internal solitons in the ocean and their effect on underwater sound, J. Acoust. Soc. Am. 121, 695 (2007).
  23. K. R. Helfrich and W. K. Melville, Long nonlinear internal waves, Annu. Rev. Fluid Mech. 38, 395 (2006).
  24. S. Watanabe, Ion acoustic soliton in plasma with negative ion, J. Phys. Soc. Jpn. 53, 950 (1984).
  25. M. S. Ruderman, T. Talipova, and E. Pelinovsky, Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions, J. Plasma Phys. 74, 639 (2008).
  26. E. Demler and A. Maltsev, Semiclassical solitons in strongly correlated systems of ultracold bosonic atoms in optical lattices, Ann. Phys. (NY) 326, 1775 (2011).
  27. A. M. Kamchatnov, Y.-H. Kuo, T.-C. Lin, T.-L. Horng, S.-C. Gou, R. Clift, G. A. El, and R. H. J. Grimshaw, Undular bore theory for the Gardner equation, Phys. Rev. E 86, 036605 (2012).
  28. A. M. Kamchatnov, Y.-H. Kuo, T.-C. Lin, T.-L. Horng, S.-C. Gou, R. Clift, G. A. El, and R. H. J. Grimshaw, Transcritical flow of a stratified fluid over topography: analysis of the forced Gardner equation, J. Fluid Mech. 736, 495 (2013).
  29. T. Kawahara, Oscillatory solitary waves in dispersive media, J. Phys. Soc. Jpn. 33, 260 (1972).
  30. P. Sprenger and M. A. Hoefer, Shock waves in dispersive hydrodynamics with nonconvex dispersion, SIAM J. Appl. Math. 77, 26 (2017).
  31. G. A. El and N. F. Smyth, Radiating dispersive shock waves in non-local optical media, Proc. R. Soc. London A 472, 20150633 (2016).
  32. S. Baqer and N. F. Smyth, Modulation theory and resonant regimes for dispersive shock waves in nematic liquid crystals, Physica D 403, 132334 (2020).
  33. Y. Pomeau, A. Ramani, and B. Grammaticos, Structural stability of the Korteweg–de Vries solitions under a singular perturbation, Physica D 31, 127 (1988).
  34. E. Benilov, R. Grimshaw, and E. Kuznetsova, The generation of radiating waves in a singularly-perturbed Korteweg–de Vries equation, Physica D 69, 270 (1993).
  35. R. Grimshaw, B. A. Malomed, and E. S. Benilov, Solitary waves with damped oscillatory tails: An analysis of the fifth-order Korteweg–de Vries equation, Physica D 77, 473 (1994).
  36. V. I. Karpman, Radiation by weakly nonlinear shallow-water solitons due to higher-order dispersion, Phys. Rev. E 58, 5070 (1998).
  37. Y. Tan, J. Yang, and D. E. Pelinovsky, Semi-stability of embedded solitons in the general fifth-order KdV equation, Wave Motion 36, 241 (2002).
  38. G. A. El and M. A. Hoefer, Dispersive shock waves and modulation theory, Physica D 333, 11 (2016).
  39. M. A. Hoefer, N. F. Smyth, and P. Sprenger, Modulation theory solution for nonlinearly resonant, fifth-order Korteweg–de Vries, nonclassical, travelling dispersive shock waves, Stud. Appl. Math. 142, 219 (2019).
  40. N. F. Smyth, Dispersive shock waves in nematic liquid crystals, Physica D 333, 301 (2016).
  41. B. B. Kadomtsev and V. I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15, 539 (1970).
  42. M. J. Ablowitz and H. Segur, On the evolution of packets of water waves, J. Fluid Mech. 92, 691 (1979).
  43. R. S. Johnson, On the inverse scattering transform, the cylindrical Korteweg–de Vries equation and similarity solutions, Phys. Lett. A 72, 197 (1979).
  44. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, New York, 2003).
  45. G. Assanto, Nematicons: Spatial Optical Solitons in Nematic Liquid Crystals (Wiley-Blackwell, New York, 2012).
  46. M. Peccianti and G. Assanto, Nematicons, Phys. Rep. 516, 147 (2012).
  47. R. S. Johnson, A Modern Introduction to the Mathematical Theory of Water Waves (Cambridge University Press, Cambridge, 1997).
  48. T. R. Marchant and N. F. Smyth, Soliton interaction for the extended Korteweg–de Vries equation, IMA J. Appl. Math. 56, 157 (1996).
  49. R. Hirota, Exact solutions to the equation describing cylindrical solitons, Phys. Lett. A 71, 393 (1979).
  50. M. D. Kruskal and H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math. 85, 129 (1991).
  51. A. C. Aitken, On Bernoulli's numerical solution of algebraic equations, Proc. R. Soc. Edinburgh 46, 289 (1926).
  52. B. Fornberg and G. B. Whitham, Numerical and theoretical study of certain non-linear wave phenomena, Philos. Trans. R. Soc. London A 289, 373 (1978).
  53. T. F. Chan and T. Kerkhoven, Fourier methods with extended stability intervals for KdV, SIAM J. Numer. Anal. 22, 441 (1985).
  54. L. N. Trefethen, Spectral Methods in MATLAB (SIAM, Philadephia, 2000).
  55. J. J. Lee, J. E. Skjelbreia, and F. Raichlen, Measurement of velocities in solitary waves, J. Water Port Coast. Ocean Div. 108, 200 (1982).
  56. H. C. Hsu, Y. Y. Chen, and H. H. Hwung, Experimental study of the particle paths in solitary water waves, Philos. Trans. R. Soc. A 370, 1629 (2012).
  57. S. Baqer, D. J. Frantzeskakis, T. P. Horikis, C. Houdeville, T. R. Marchant, and N. F. Smyth, Nematic dispersive shock waves from nonlocal to local, Appl. Sci. 11, 4736 (2021).
  58. G. Assanto and N. F. Smyth, Self-confined light waves in nematic liquid crystals, Physica D 402, 132182 (2020).
  59. A. Piccardi, A. Alberucci, N. Tabiryan, and G. Assanto, Dark nematicons, Opt. Lett. 36, 1356 (2011).
  60. N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, Shocks in Nonlinear Media, Phys. Rev. Lett. 99, 043903 (2007).
  61. T. P. Horikis and D. J. Frantzeskakis, Ring dark and antidark solitons in nonlocal media, Opt. Lett. 41, 583 (2016).
  62. T. P. Horikis and D. J. Frantzeskakis, Asymptotic reductions and solitons of nonlocal nonlinear Schrödinger equations, J. Phys. A: Math. Theor. 49, 205202 (2016).

Outline

Information

Phys. Rev. Fluids 6, 104401– Published 25 October, 2021
  • Received 12 July 2021
  • Accepted 7 October 2021
Crossmark - Check for updates button

©2021 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation