Second-order perturbation theory for the parametrically driven NLS solitons

The perturbation theory based on the Riemann-Hilbert problem is constructed for the nonlinear Schrödinger equation. The second-order equations for the spectral data describing the soliton-wave interaction are obtained. The theory is applied to the parametrically driven, damped nonlinear Schrödinger equation. The parametrically driven NLS soliton is shown to become unstable, when the driving strength is greater then the critical value, due to the resonant interaction with the linear waves. 03.40.KfWaves and wave propagation: general mathematical aspects Typeset using REVTEX 1