Parametric integration by Laks of nonlinear dynamic systems and problem of splitting of multisoliton separatrix varieties

Abstract

A new analytic approach to the description of the class of parametrically Lax-integrable nonlinear nonhomogeneous dynamical systems defined on functional manifolds that generalizes the well-known Mitropol'skii asymptotic method [1] is developed. Under a stipulated $\varepsilon$-deformation of the initial dynamical system, $\varepsilon\rightarrow0$, the bifurcation problem for multisoliton separatrix manifolds is studied on the basis of the concept of a generalized Mitropol'skii-Mel'nikov $\mu$-function. In the special case of a nonlinear Korteweg -de Vries-Bürgers dynamical system, the structure of the bifurcation of a homoclinic separatrix trajectory is studied as a function of the embedding parameters of a soliton manifold in a functional space.