Research output: Working paper › Working paper and Preprints

Original language | English |
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Publication date | 2001 |

State | Unpublished |

`Embedded solitons' is a name given to localised solutions of multi-component dispersive wave models, that occur in resonance with the linear spectrum. They exist at isolated (codimension-one) parameter values within the steady state equations which have linearisation with eigenvalues ±lambda and ±i omega. At the same time, quasi-solitons, or generalised solitary waves with non-decaying radiation tails, are known to be endemic. We consider the general question of when two quasi-solitons may be glued together to form a two-humped `bound state' embedded soliton in the limit lambda/omega goes to 0. A generalisation of the method of Gorshkov and Ostrovsky is used within the framework of general normal forms for Hamiltonian reversible systems. A simple asymptotic formula is derived that governs the existence, symmetry and accumulation rate of the bound states. This generalises earlier ad hoc calculations for specific examples. The formula depends on a few simple ingredients: the eigenvalues of the equilibrium, an asymptotic estimate for the tail amplitude of the quasi-solitons and the `Birkhoff signature' and symmetry properties of the Hamiltonian. Only the tail amplitude estimate is difficult to calculate, requiring exponential asymptotics. But it is shown that this only affects the third-order term in the asymptotic formula. The calculation is worked out in detail for the steady states of several example PDE systems, taken from nonlinear optics and fluid mechanics; and excellent agreement with numerical results is found.