Generation of
ocean waves in near-surface wind currents and
modulation of internal waves have been
studied in physical oceanography in context of remote
sensing of the ocean surface. New analytical
integral-differential models serve as simple tools
to describe the nonlinear dynamics of the ocean fluid.
Analysis of short-wave packets in deep stratified fluid is based on a nonlocal version of the nonlinear Schrodinger equation (the intermediate NLS equation). This integral - differential equation is a reduction of another integrable nonlinear equation, the Benjamin - Ono equation for internal waves. Therefore, it is no surprise to find out that the intermediate NLS equation inherits all integrability properties from the parent equation.
Analysis of long internal waves in uniformly stratified fluid is based on the Korteweg-de Vries equation with an integral nonlinear time-dependent evolution term. Asymptotic estimates can be deduced from this model to relate the wave instabilities in inviscid stratified fluids with wave breaking and vortex formation.
Analysis of resonant near-surface shear flows with internal waves in shear stratified fluid is based on the system of coupled Kortewed-de Vries equations. This system can be transformed to an integral Whitham model for long dispersive waves. Full integration of the model for stationary solutions with constant propagation speed shows existence of critical solitons with sharp corners (peakons).
