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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
Representation in Riemann invariants [Baines, 1995; Lyapidevsky & Teshukov 2000; Slunyaev et al, 2003] 2-layer fluid rigid-lid boundary condition Boussinesq approximation
The fully nonlinear (but dispersiveless) model The full nonlinear velocity [Slunyaev et al, 2003; Grue & Ostrovsky, 2003]
The full nonlinear velocity
1 1 2 2 u1u1 u1u1 u2u2 u2u2 c lin V+V+ V+V+ Velocity profiles h = 0.1 h = 0.5
The full nonlinear velocity asymptotic expansions for any-order nonlinear coefficients
etc… The full nonlinear velocity
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
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
Rigorous way for obtaining asymptotic eqs stratified fluid free surface condition
Rigorous way for obtaining asymptotic eqs stratified fluid free surface condition extGE
Asymptotical integrability Asymptotical integrability (Marchant&Smyth, Fokas&Liu 1996) 2nd order KdV KdV
Almost asymptotical integrability GE extGE
Almost asymptotical integrability GE extGE
Almost asymptotical integrability GE extGE
2-order GE theory as perturbations of the GE solutions Qualitative closeness of the GE and its extensions
Initial Problem AKNS approach
GE AKNS approach mKdV AKNS approach
GE mKdV AKNS approach
GE mKdV a – is an arbitrary number
GE Passing through a turning point? t Tasks:
GE Passing through a turning point? t Tasks: A solitary-like wave over a long-scale wave
GE A solitary-like wave over a long-scale wave
GE+ mKdV+ a soliton cannot pass through a too high wave being a soliton discrete eigenvalues may become continuous a
GE+ mKdV+ soliton amplitude ( s denotes polarity) soliton velocity Solitons
GE- mKdV- at the turning point all spectrum becomes continuous
GE- mKdV- soliton amplitude soliton velocity
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
Spatio-temporal evolution NLS breather envelope soliton This approach was applied to the NLS eq
Solitary wave dynamics on pedestals may be interpreted Strong change of waves may be predicted (turning points)
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