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D.E. Pelinovsky and J. Yang

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Stability analysis of embedded solitons
in the generalized third-order NLS equation

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Chaos 15, 037115 (2005)

**Abstract:**

We study the generalized third-order NLS equation which admits a
one-parameter family of single-hump embedded solitons. Analyzing the
spectrum of the linearization operator near the embedded soliton, we
show that there exists a resonance pole in the left half-plane of
the spectral parameter, which explains linear stability, rather than
nonlinear semi-stability, of embedded solitons. Using exponentially
weighted spaces, we approximate the resonance pole both analytically
and numerically. We confirm in a near-integrable asymptotic limit
that the resonance pole gives precisely the linear decay rate of
parameters of the embedded soliton. Using conserved quantities, we
qualitatively characterize the stable dynamics of embedded solitons.

**Keywords:**

THIRD-ORDER DERIVATIVE NONLINEAR SCHRODINGER EQUATIONS,
EMBEDDED SOLITONS, SPECTRAL STABILITY, EIGENVALUES AND RESONANCE POLES,
ASYMPTOTIC APPROXIMATIONS OF EIGENVALUES