Deep-water “limiting” envelope solitons

1. Deep-water “limiting” envelope solitons Alexey Slunyaev Institute of Applied Physics RAS, Nizhny Novgorod

2. Motivation

4. NLS envelope solitons two collision of solitons ka = 0.085 [Zakharov et al, 2006]

5. Limiting envelope solitons appearance and propagation of “limiting” envelope solitons (“breathers”) steepness profile [Dyachenko & Zakharov, 2008]

6. ? How do the approximate (high-order) envelope eqs and fully nonlinear description of steep envelope solitary waves relate?

7. Brief overview of the history Water wave envelope solitons Envelope solitons results from modulational instability (~1965) Envelope solitons are the asymptotic solution of NLS (1968, 1971, 1973) Collision of envelope solitary waves Longuet-Higgins & Phillips, JFM 1962 (analytics) Feir, Proc R Soc A 1965 (experiment) Zakharov & Shabat, JETP 1971, 1973 (integrability) Dommermuth & Yue, JFM 1987 (HOSM) West et al, JGR 1987 (HOSM) Zakharov et al, Eur J Mech B Fl 2006 (full eds)

8. Models

9. Full equations for potential gravity surface waves

10. Full numerical model incompressible inviscid irrotational water potential movement gravity force infinite depth periodic boundary conditions Euler eqs in conformal variables [Zakharov et al, 2002] High-Order Spectral Method (HOSM), M = 6 [Dommermuth&Yue, West et al, 1987]

11. Envelope equation

12. Envelope equation

13. Envelope equation

14. Envelope equation

15. Envelope equation

16. Choosing the approximate model Modulation equations Classic NLS Soliton solution NLS-2 Dysthe or MNLS

17. Approximate model , Free and bound waves Example of a laboratory frequency spectrum of intense narrow-banded wave groups . Bound wave 3 order corrections

18. Propagation of single envelope solitons over deep water

19. Single envelope solitons Initial condition Exact solution of the NLS soliton Bound wave correction

20. Single envelope solitons Initial condition Nonlinearity / dispersion ration in the NLS eq «Nonlinear» time

21. Propagation of single solitons Surface displacements ka = 0.2, T0 = 2 Tnl 50 Full Dysthe ka = 0.3, T0 = 2 Tnl 20 Full Dysthe

22. Propagation of single solitons Characteristic steepness & max wave slope k [max((x)) – min((x))] / 2 max|x| ka = 0.2, T0 = 2 Tnl 50 Full Dysthe ka = 0.3, T0 = 2 Tnl 20 Full Dysthe

23. Propagation of single solitons Role of high-order corrections Accuracy Model Eq Field reconstruction Soliton solution Classic NLS O(3) 1 O(3) Dysthe O(3+1) 1 + 2 O(3+1) We use O(3+1) 1 + 2 +3 O(3)

24. Propagation of single solitons Role of high-order corrections ka = 0.3, T0 = 2 3-order bound wave corrections Full Dysthe no bound wave corrections Full Dysthe

25. Envelope solitoninteractions

26. Soliton interaction Toward propagation a1 = 0.2, a2 = 0.2k1 = 1, k2 = 1k1a1 = 0.2, k2a2 = 0.2 a1 = 0.2, a2 = 0.1k1 = 1, k2 = 1k1a1 = 0.2, k2a2 = 0.1 a1 = 0.2, a2 = 0.1k1 = 1, k2 = 2k1a1 = 0.2, k2a2 = 0.2

27. Soliton interaction Toward propagation half-height & slope

28. Soliton interaction Toward propagation after 7 (6) collisions

29. Soliton interaction Comoving propagation a1 = 0.1, a2 = 0.1k1 = 1, k2 = 2k1a1 = 0.1, k2a2 = 0.2 a1 = 0.05, a2 = 0.1k1 = 1, k2 = 2k1a1 = 0.05, k2a2 = 0.2 a1 = 0.2, a2 = 0.1k1 = 1, k2 = 2k1a1 = 0.2, k2a2 = 0.2

30. Soliton interaction Comoving propagation half-height & slope

31. Soliton interaction Comoving propagation after one collision

32. Soliton interaction Bi-soliton Exact 2-soliton solution of NLS + bound wave corrections a1 = 0.2, a2 = 0.1 k = 1ka1 = 0.1, ka2 = 0.2

33. Soliton interaction Bi-soliton Full Tnl 50 Dysthe

34. Soliton interaction Bi-soliton Coupled nonlinear groups Full a1 = 0.08, a2 = 0.04 k = 1ka1 = 0.08, ka2 = 0.04 Dysthe overlapping solitons & not exact solution 100Tnl background noise long-time simulation non-Hamiltonian Dysthe eq 400Tnl

35. Conclusions Existence of “limiting” envelope solitons has been shown [Dyachenko & Zakharov, 2008] High-order envelope models describe the “limiting” envelope solitons (up to ka~ 0.2...0.3) quite well Occasional wave steepening seems to be the only reason why the “limiting” envelope solitons are difficult to reproduce Toward-propagating envelope solitons collide in a great extent elastically When co-moving solitons interact, a higher and longer-wavelength soliton destroys the smaller and shorter-wavelength one Long-time interacting solitons (with similar wavenumbers) can couple

36. Thank you for your attention!