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ANALYTIC SOLUTIONS FOR LONG INTERNAL WAVE MODELS WITH IMPROVED NONLINEARITY

ANALYTIC SOLUTIONS FOR LONG INTERNAL WAVE MODELS WITH IMPROVED NONLINEARITY. Alexey Slunyaev Insitute of Applied Physics RAS Nizhny Novgorod, Russia. 2-layer fluid rigid-lid boundary condition Boussinesq approximation. 1. 2. Representation in Riemann invariants.

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ANALYTIC SOLUTIONS FOR LONG INTERNAL WAVE MODELS WITH IMPROVED NONLINEARITY

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  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 conditionBoussinesq approximation

  3. 1 2

  4. Representation inRiemann invariants [Baines, 1995;Lyapidevsky & Teshukov 2000;Slunyaev et al, 2003] 2-layer fluid rigid-lid boundary conditionBoussinesq 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. Velocity profiles h = 0.5 h = 0.1 clin u1 clin u2 u2 2 V+ 1 2 V+ u1 1

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

  9. The full nonlinear velocity etc…

  10. 2-layer fluid rigid-lid boundary conditionBoussinesq approximation The full nonlinear velocity Exact relation for H1 = H2 Corresponds to the Gardner eq

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

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

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

  15. GE Almost asymptotical integrability extGE

  16. GE Almost asymptotical integrability extGE

  17. GE Almost asymptotical integrability 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

  21. GE

  22. GE Initial Problem AKNS approach

  23. GE AKNS approach mKdV AKNS approach

  24. GE mKdV

  25. GE AKNS approach mKdV

  26. mKdV GE a – is an arbitrary number

  27. GE Tasks: Passing through a turning point?  =  (t)

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

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

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

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

  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 The initial conditions: an envelope soliton and a plane wave background periodical boundary conditions periodical boundary conditions an envelope soliton plane wave plane wave

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

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

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

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