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 [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

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

Solitary waves

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

GE

Initial Problem AKNS approach

GE AKNS approach mKdV AKNS approach

GE mKdV

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)

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