Multi-pulse embedded solitons as bound states of quasi-solitons

Abstract

‘Embedded solitons’ (ESs) 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 ± λ and ±i ω. At the same time, quasi-solitons (QSs), or generalised solitary waves with non-decaying radiation tails, are known to be endemic. We consider the general question of when two QSs may be glued together to form a two-humped ‘bound state’ (BS) ES in the limit λ/ ω→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 BSs. This generalises earlier ad hoc calculations for specific examples. The formula depends on few simple ingredients: the eigenvalues of the equilibrium, an asymptotic estimate for the tail amplitude of the QSs, 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.