1. ANALYTIC SOLUTIONS FOR LONG INTERNAL WAVE MODELS WITH IMPROVED NONLINEARITY Alexey Slunyaev Insitute of Applied Physics RAS Nizhny Novgorod, Russia

2. 2-layer fluid rigid-lid boundary condition Boussinesq approximation

3. 1 2

4. Representation in Riemann invariants [Baines, 1995; Lyapidevsky & Teshukov 2000; Slunyaev et al, 2003] 2-layer fluid rigid-lid boundary condition Boussinesq approximation

5. The fully nonlinear (but dispersiveless) model The full nonlinear velocity [Slunyaev et al, 2003; Grue & Ostrovsky, 2003]

6. The full nonlinear velocity

7. 1 1 2 2 u1u1 u1u1 u2u2 u2u2 c lin V+V+ V+V+ Velocity profiles h = 0.1 h = 0.5

8. The full nonlinear velocity asymptotic expansions for any-order nonlinear coefficients

9. etc… The full nonlinear velocity

10. Exact relation for H 1 = H 2 The full nonlinear velocity Corresponds to the Gardner eq 2-layer fluid rigid-lid boundary condition Boussinesq approximation

11. Exact fully nonlinear velocity for asymp eqs Exact velocity fields (hydraulic approx) Strongly nonlinear wave steepening (dispersionless approx) The GE is exact when the layers have equal depths

12. Rigorous way for obtaining asymptotic eqs stratified fluid free surface condition

13. Rigorous way for obtaining asymptotic eqs stratified fluid free surface condition extGE

14. Asymptotical integrability Asymptotical integrability (Marchant&Smyth, Fokas&Liu 1996) 2nd order KdV KdV

15. Almost asymptotical integrability GE extGE

16. Almost asymptotical integrability GE extGE

17. Almost asymptotical integrability GE extGE

18. Solitary waves

19. 2-order GE theory as perturbations of the GE solutions Qualitative closeness of the GE and its extensions

20. GE

22. Initial Problem AKNS approach

23. GE AKNS approach mKdV AKNS approach

24. GE mKdV

25. GE mKdV AKNS approach

26. GE mKdV a – is an arbitrary number

27. GE Passing through a turning point? t Tasks:

28. GE Passing through a turning point? t Tasks: A solitary-like wave over a long-scale wave

29. GE A solitary-like wave over a long-scale wave

30. GE+ mKdV+ a soliton cannot pass through a too high wave being a soliton discrete eigenvalues may become continuous a

31. GE+ mKdV+ soliton amplitude ( s denotes polarity) soliton velocity Solitons

32. GE- mKdV- at the turning point all spectrum becomes continuous

33. GE- mKdV- soliton amplitude soliton velocity

34. This approach was applied to the NLS eq periodical boundary conditions an envelope soliton plane wave The initial conditions: an envelope soliton and a plane wave background

35. Spatio-temporal evolution NLS breather envelope soliton This approach was applied to the NLS eq

36. Solitary wave dynamics on pedestals may be interpreted Strong change of waves may be predicted (turning points)

37. Thank you for attention! Gavrilyuk S. Grimshaw R. Pelinovsky E. Pelinovsky D. Polukhina O. Talipova T. Co-authors