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D.E. Pelinovsky and J. Yang

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A normal form for nonlinear resonance of embedded solitons

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Proc. R. Soc. Lond. A 458, 1469-1497 (2002)

**Abstract:**

A normal form for nonlinear resonance of embedded solitons is derived
for a coupled two-wave system that generalizes the second-harmonic-generating
model. This wave system is non-Hamiltonian in general. An embedded soliton
is a localized mode of the nonlinear system that coexists with the linear
wave spectrum. It occurs as a result of a co-dimension one bifurcation of
nonlocal wave solutions. Nonlinearity couples the embedded soliton and the
linear wave spectrum and induces a one-sided radiation-driven decay of
embedded solitons. The normal form shows that the embedded soliton is
semi-stable, i.e., it survives under perturbations of one sign, but is
destroyed by perturbations of the opposite sign. When a perturbed embedded
soliton sheds continuous wave radiation, the radiation amplitude is generally
not minimal even if the wave system is Hamiltonian. The results of the
analytical theory are confirmed by numerical computations.

**Keywords:**

EMBEDDED SOLITONS, SECOND-HARMONIC-GENERATING WAVE SYSTEM, NONLINEAR RESONANCE,
NORMAL FORM, RADIATION-DRIVEN SEMI-STABILITY, SPECTRAL ANALYSIS