D.E. Pelinovsky and J. Yang

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

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