An approximate orthogonal decomposition method for the solution of the generalized liouville equation

Abstract

We consider an approximate integration method of the Cauchy problem for the generalized Liouville equation using symbolic and numeric computer computations. This method is based on the probability density function orthonormal series expansion in the small and initial time space domains. We are investigating several expansions and determine their convergence conditions to ensure the convergence of the asymptotic expansion to the solution of the considered problem.To illustrate the applicability of the introduced asymptotic orthogonal decompositions [18] we took the describing bidimensional integrable dispersive shallow water equation developed by Roberto Camassa and Darryl D. Holm, Los Alamos National Laboratory. Since CH-equation solutionsare represented by a superposition of arbitrary number of peakons (peaked solitons) [9],[16], one can compare the coincidence of the \peakon" solutions character provided by numerical modeling along some trajectories for truncated asymptotic series expansions obtained by symbolic computations.