Yury Stepanyants
School of Mathematics, Physics and Computing, University of Southern Queensland,
Toowoomba, Australia
Exact and asymptotic methods are presented for the description of the interaction of plane and fully two-dimensional solitary waves in non-linear dispersive media. The asymptotic approach allows one to describe stationary wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory are validated by comparison with the exact two-soliton solutions of the completely integrable Kadomtsev–Petviashvili equation. The suggested approach is equally applicable to a wide class of non-integrable equations too. Two examples of application of the asymptotic theory are presented. The first one is the description of plane soliton interactions in the infinitely deep stratified ocean governed by the 2D Benjamin–Ono equation. The second example pertains to the description of plane soliton interactions in 2D discrete lattices of quadratic or triangle-hexagonal structures.
In the second part of this talk, the dynamics of fully localized solitary waves called lumps is presented. Lumps can exist in nonlinear media with positive dispersion. It is shown that lumps can form molecular-type patterns (multi-lumps) stationary moving in a certain direction on a plane. Various analytical and numerical methods of investigation of lump interaction are presented. Elementary acts of lump interaction with each and with plane solitons are described. Nontrivial interactions of lumps and multi-lumps with each other are illustrated through numerical modelling. Interaction of periodic lump chains with plane solitons is also presented