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

2. Motivation

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

5. appearance and propagation of “limiting” envelope solitons (“breathers”) steepness profile Limiting envelope solitons [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) 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) Feir, Proc R Soc A 1965 (experiment)

8. Models

9. Full equations for potential gravity surface waves

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

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

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

21. Surface displacements Propagation of single solitons ka = 0.2, T 0 = 2  ka = 0.3, T 0 = 2  Full Dysthe T nl  50 T nl  50 T nl  20 T nl  20

22. Characteristic steepness & max wave slope Propagation of single solitons ka = 0.2, T 0 = 2  ka = 0.3, T 0 = 2  Full Dysthe T nl  50 T nl  50 T nl  20 T nl  20 max|  x | k [max(  (x)) – min (  (x))] / 2

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

24. Role of high-order corrections Propagation of single solitons ka = 0.3, T 0 = 2  3-order bound wave corrections no bound wave corrections Full Dysthe Full Dysthe

25. Envelope soliton interactions

26. Soliton interaction Toward propagation a 1 = 0.2, a 2 = 0.2 k 1 = 1, k 2 = 1 k 1 a 1 = 0.2, k 2 a 2 = 0.2 a 1 = 0.2, a 2 = 0.1 k 1 = 1, k 2 = 1 k 1 a 1 = 0.2, k 2 a 2 = 0.1 a 1 = 0.2, a 2 = 0.1 k 1 = 1, k 2 = 2 k 1 a 1 = 0.2, k 2 a 2 = 0.2

27. Soliton interaction Toward propagation half-height & slope

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

29. Soliton interaction Comoving propagation a 1 = 0.2, a 2 = 0.1 k 1 = 1, k 2 = 2 k 1 a 1 = 0.2, k 2 a 2 = 0.2 a 1 = 0.1, a 2 = 0.1 k 1 = 1, k 2 = 2 k 1 a 1 = 0.1, k 2 a 2 = 0.2 a 1 = 0.05, a 2 = 0.1 k 1 = 1, k 2 = 2 k 1 a 1 = 0.05, k 2 a 2 = 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 a 1 = 0.2, a 2 = 0.1 k = 1 ka 1 = 0.1, ka 2 = 0.2

33. Soliton interaction Bi-soliton Full Dysthe T nl  50 T nl  50

34. Soliton interaction Bi-soliton Coupled nonlinear groups overlapping solitons & not exact solution background noise non-Hamiltonian Dysthe eq long-time simulation  400T nl  400T nl  100T nl  100T nl Full Dysthe a 1 = 0.08, a 2 = 0.04 k = 1 ka 1 = 0.08, ka 2 = 0.04

35. Conclusions Existence of “limiting” envelope solitons has been shown [Dyachenko & Zakharov, 2008] Occasional wave steepening seems to be the only reason why the “limiting” envelope solitons are difficult to reproduce High-order envelope models describe the “limiting” envelope solitons (up to ka ~ 0.2...0.3) quite well 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!