Applied Mathematics Seminar

Embedded solitons are solitary wave solutions of coupled or higher-order nonlinear wave equations which occur despite the co-existence of linear waves. Example include 5th-order or coupled 3rd-order KdV equations arising in fluid mechanics and coupled NLS equations arising in nonlinear optics. This talk shall primarily review what is known about their existence properties using arguments from dynamical systems theory. It will be shown that the solutions are typically isolated, as one parameter is needed to kill the amplitude of the radiation in the tail of the solitary wave. In the singular limit of fast radiation (or weakly localised waves) this leads to a problem that is beyond all orders. Two methods of cancelling the tail radiation will be considered, to produce either one-humped or multi-humped waves. Using numerical continuation, the structure and multiplicity of embedded solitons will be uncovered for several examples. Finally computational results will be reviewed showing that these solitary waves are at best neutrally stable as solutions of the initial-value problem and are generically subject to a one-sided algebraic instability.