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

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2-layer fluid rigid-lid boundary condition Boussinesq approximation

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1 2

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

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

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The full nonlinear velocity

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

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

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etc… The full nonlinear velocity

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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

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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

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Rigorous way for obtaining asymptotic eqs stratified fluid free surface condition

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

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

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Almost asymptotical integrability GE extGE

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Almost asymptotical integrability GE extGE

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Almost asymptotical integrability GE extGE

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Solitary waves

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

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GE

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Initial Problem AKNS approach

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GE AKNS approach mKdV AKNS approach

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GE mKdV

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GE mKdV AKNS approach

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GE mKdV a – is an arbitrary number

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GE Passing through a turning point? t Tasks:

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

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

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

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

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

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GE- mKdV- soliton amplitude soliton velocity

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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

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

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

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

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