A multidimensional superposition principle: numerical simulation and analysis of soliton invariant manifolds I

In the framework of a multidimensional superposition principle involving an analytical approach to nonlinear PDEs, a numerical technique for the analysis of soliton invariant manifolds is developed. This experimental methodology is based on the use of computer simulation data of soliton-perturbation interactions in a system under investigation, and it allows the determination of the dimensionality of similar manifolds and partially (in the small amplitude perturbation limit) to restore the related superposition formulae. Its application for cases of infinite dimensionality,and the question of approximation by lower dimensional manifolds and, respectively, by superposition formulae of a lower order are considered as well. The ideas and implementation details are illustrated and verified by using examples with the integrable, MKdV and KdV equations, and also nonintegrable, Kawahara and Regularized Long Waves equation, soliton models.