Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2010-07-19
Nonlinear Sciences
Exactly Solvable and Integrable Systems
19 pages
Scientific paper
Special polynomials play a role in several aspects of soliton dynamics. These are differential polynomials in u, the solution of a nonlinear evolution equation, which vanish identically when u represents a single soliton. Local special polynomials contain only powers of u and its spatial derivatives. Non-local special polynomials contain, in addition, non-local entities (e.g., \delta x-1u). When u is a multiple-solitons solution, local special polynomials are localized in the vicinity of the soliton-collision region and fall off exponentially in all directions away from this region. Non-local ones are localized along soliton trajectories. Examples are presented of how, with the aid of local special polynomials, one can modify equations that have only a single-soliton solution into ones, which have that solution as well as, at least, a two-solitons solutions. Given an integrable equation, with the aid of local special polynomials, it is possible to find all evolution equations in higher scaling weights, which share the same single-soliton solution and are either integrable, or, at least, have a two-solitons solution. This is demonstrated for one or two consecutive scaling weights for a number of known equations. In the study of perturbed integrable equations, local special polynomials are responsible for inelastic soliton interactions generated by the perturbation in the multiple-soliton case, and for the (possible) loss of asymptotic integrability. Non-local special polynomials describe higher-order corrections to the solution, which are of an inelastic nature.
No associations
LandOfFree
Special polynomials and soliton dynamics does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Special polynomials and soliton dynamics, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Special polynomials and soliton dynamics will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-63099