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N.V. Alexeeva, I.V. Barashenkov, and D.E. Pelinovsky

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Dynamics of the parametrically driven NLS solitons
beyond the onset of the oscillatory instability

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Nonlinearity 12, 103-140 (1999)

**Abstract:**

Solitary waves in conservative and near-conservative systems
may become unstable due to a resonance of two internal oscillation
modes. We study the parametrically
driven, damped nonlinear Schrodinger equation, a prototype system
exhibiting this oscillatory instability An asymptotic multi-scale
expansion is used to derive a
reduced amplitude equation describing the nonlinear stage of the
instability and supercritical dynamics of the soliton in the
weakly dissipative case. We also derive the
amplitude equation in the strongly dissipative case, when the
bifurcation is of the Hopf type. The analysis of the reduced
equations shows that in the undamped case the
temporally periodic spatially localized structures are suppressed
by the nonlinearity-induced radiation. In this case the unstable
stationary soliton evolves either into a
slowly decaying long-lived breather, or into a radiating soliton
whose amplitude grows without bound. However, adding a small
damping is sufficient to bring about a
stably oscillating soliton of finite amplitude.

**Keywords**:

NONLINEAR SCHRODINGER EQUATION, SINE-GORDON EQUATION,
PHASE-SENSITIVE AMPLIFICATION, EXCITED SOLITARY WAVES,
FARADAY RESONANCE, STABILITY, CHAOS, BREATHER, ATTRACTORS