THE ONSET OF STOCHASTIC PULSATIONS AT THE EARLY NONLINEAR STAGE OF THE DEVELOPMENT OF PERTURBATIONS IN THE JET OF AN INCOMPRESSIBLE FLUID PROPAGATING ALONG A WALL
Aa. Burov et Os. Ryzhov, THE ONSET OF STOCHASTIC PULSATIONS AT THE EARLY NONLINEAR STAGE OF THE DEVELOPMENT OF PERTURBATIONS IN THE JET OF AN INCOMPRESSIBLE FLUID PROPAGATING ALONG A WALL, Journal of applied mathematics and mechanics, 56(6), 1992, pp. 921-927
The Korteweg-De Vries equation, which describes the non-linear propaga
tion 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 equat
ion becomes inhomogeneous. If an external action is specified in the f
orm of a running wave, the particular solution of the inhomogeneous eq
uation may be sought in an analogous form. As a result, the simplest p
roblem in the theory of dynamical systems in the Hamiltonian formulati
on arises. As usual, the existence of a homoclinic structure in the ne
ighbourhood of the separatrices is deduced from an analysis of a Poinc
are transformation. Among the trajectories belonging to the homoclinic
structure in the secant plane, there are some with properties which a
re formulated in terms of determinate chaos. A fundamentally important
conclusion concerning the dual role of solitons at the non-linear sta
ge of the wave motion of the fluid follows: on the one hand, they serv
e as the nuclei of large-scale coherent structures and. on the other h
and, they are responsible for the onset of stochastic pulsations.