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On the theory of internal waves of permanent form in fluids of great depth. (English) Zbl 0829.76012
A stationary solitary wave moving along a border between two layers of an ideal liquid with different densities is considered. The lower layer has a finite depth, while the upper one is assumed infinitely deep. The horizontal velocity of the liquid at infinity (in the vertical direction) is treated as an unknown parameter that must be found together with the shape of the interface corresponding to a stationary internal solitary wave. The problem is formulated on the basis of equations for the stream function, supplemented by the boundary conditions at the interface. By means of the hodograph transformation, the problem is mapped into a Laplace equation for an infinite strip with the Dirichlet boundary condition at the bottom, and a nonlocal nonlinear boundary condition at the top. Subsequent analysis allows one to reduce the latter problem, in the case when the above-mentioned horizontal velocity at infinity takes values slightly larger than the critical value at which a bifurcation of small solutions occurs, to the stationary Benjamin-Ono (BO) equation with a small correction. It is demonstrated that, as in the case of the pure BO equation, the solitary wave solution is given by a rational rather than by an exponential function.

MSC:
76B25 Solitary waves for incompressible inviscid fluids
76B55 Internal waves for incompressible inviscid fluids
76V05 Reaction effects in flows
35Q51 Soliton equations
35J25 Boundary value problems for second-order elliptic equations
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