The onset of stochastic pulsations at the early non-linear stage of the development of perturbations in the jet of an incompressible fluid propagating along a wall

The Korteweg-De Vries equation, which describes the non-linear propagation of perturbations in a jet of incompressible fluid emanating from a slit in a planar screen and propagating along a wall is considered. When account is taken of the natural vibrations of the wall, the equation becomes inhomogeneous. If an external action is specified in the form of a running wave, the particular solution of the inhomogeneous equation may be sought in an analogous form. As a result, the simplest problem in the theory of dynamical systems in the Hamiltonian formulation arises. As usual, the existence of a homoclinic structure in the neighbourhood of the separatrices is deduced from an analysis of a Poincaré transformation. Among the trajectories belonging to the homoclinic structure in the secant plane, there are some with properties which are formulated in terms of determinate chaos. A fundamentally important conclusion concerning the dual role of solitons at the non-linear stage of the wave motion of the fluid follows: on the one hand, they serve as the nuclei of large-scale coherent structures and, on the other hand, they are responsible for the onset of stochastic pulsations.